{
"query": {
"display": "$$\\int\\:\\left(3x^{2}-7x\\right)^{5}\\left(6x-7\\right)dx$$",
"symbolab_question": "BIG_OPERATOR#\\int (3x^{2}-7x)^{5}(6x-7)dx"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Integrals",
"subTopic": "Indefinite Integrals",
"default": "\\frac{243x^{12}}{2}-1701x^{11}+\\frac{19845x^{10}}{2}-30870x^{9}+\\frac{108045x^{8}}{2}-50421x^{7}+\\frac{117649x^{6}}{6}+C",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\int\\:\\left(3x^{2}-7x\\right)^{5}\\left(6x-7\\right)dx=\\frac{243x^{12}}{2}-1701x^{11}+\\frac{19845x^{10}}{2}-30870x^{9}+\\frac{108045x^{8}}{2}-50421x^{7}+\\frac{117649x^{6}}{6}+C$$",
"input": "\\int\\:\\left(3x^{2}-7x\\right)^{5}\\left(6x-7\\right)dx",
"steps": [
{
"type": "interim",
"title": "Expand $$\\left(3x^{2}-7x\\right)^{5}\\left(6x-7\\right):{\\quad}1458x^{11}-18711x^{10}+99225x^{9}-277830x^{8}+432180x^{7}-352947x^{6}+117649x^{5}$$",
"input": "\\left(3x^{2}-7x\\right)^{5}\\left(6x-7\\right)",
"steps": [
{
"type": "interim",
"title": "$$\\left(3x^{2}-7x\\right)^{5}=243x^{10}-2835x^{9}+13230x^{8}-30870x^{7}+36015x^{6}-16807x^{5}$$",
"input": "\\left(3x^{2}-7x\\right)^{5}",
"steps": [
{
"type": "step",
"primary": "Apply binomial theorem: $$\\left(a+b\\right)^{n}=\\sum_{i=0}^{n}\\binom{n}{i}a^{\\left(n-i\\right)}b^{i}$$",
"secondary": [
"$$a=3x^{2},\\:\\:b=-7x$$"
],
"meta": {
"practiceLink": "/practice/expansion-practice#area=main&subtopic=Binomial%20Expansion",
"practiceTopic": "Binomial Expansion"
}
},
{
"type": "step",
"result": "=\\sum_{i=0}^{5}\\binom{5}{i}\\left(3x^{2}\\right)^{\\left(5-i\\right)}\\left(-7x\\right)^{i}"
},
{
"type": "interim",
"title": "Expand summation",
"result": "=\\frac{5!}{0!\\left(5-0\\right)!}\\left(3x^{2}\\right)^{5}\\left(-7x\\right)^{0}+\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\left(-7x\\right)^{1}+\\frac{5!}{2!\\left(5-2\\right)!}\\left(3x^{2}\\right)^{3}\\left(-7x\\right)^{2}+\\frac{5!}{3!\\left(5-3\\right)!}\\left(3x^{2}\\right)^{2}\\left(-7x\\right)^{3}+\\frac{5!}{4!\\left(5-4\\right)!}\\left(3x^{2}\\right)^{1}\\left(-7x\\right)^{4}+\\frac{5!}{5!\\left(5-5\\right)!}\\left(3x^{2}\\right)^{0}\\left(-7x\\right)^{5}",
"steps": [
{
"type": "step",
"primary": "$$\\binom{n}{i}=\\frac{n!}{i!\\left(n-i\\right)!}$$"
},
{
"type": "step",
"primary": "$$\\quad\\:i=0\\quad:\\quad\\frac{5!}{0!\\left(5-0\\right)!}\\left(3x^{2}\\right)^{5}\\left(-7x\\right)^{0}$$"
},
{
"type": "step",
"primary": "$$\\quad\\:i=1\\quad:\\quad\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\left(-7x\\right)^{1}$$"
},
{
"type": "step",
"primary": "$$\\quad\\:i=2\\quad:\\quad\\frac{5!}{2!\\left(5-2\\right)!}\\left(3x^{2}\\right)^{3}\\left(-7x\\right)^{2}$$"
},
{
"type": "step",
"primary": "$$\\quad\\:i=3\\quad:\\quad\\frac{5!}{3!\\left(5-3\\right)!}\\left(3x^{2}\\right)^{2}\\left(-7x\\right)^{3}$$"
},
{
"type": "step",
"primary": "$$\\quad\\:i=4\\quad:\\quad\\frac{5!}{4!\\left(5-4\\right)!}\\left(3x^{2}\\right)^{1}\\left(-7x\\right)^{4}$$"
},
{
"type": "step",
"primary": "$$\\quad\\:i=5\\quad:\\quad\\frac{5!}{5!\\left(5-5\\right)!}\\left(3x^{2}\\right)^{0}\\left(-7x\\right)^{5}$$"
},
{
"type": "step",
"result": "=\\frac{5!}{0!\\left(5-0\\right)!}\\left(3x^{2}\\right)^{5}\\left(-7x\\right)^{0}+\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\left(-7x\\right)^{1}+\\frac{5!}{2!\\left(5-2\\right)!}\\left(3x^{2}\\right)^{3}\\left(-7x\\right)^{2}+\\frac{5!}{3!\\left(5-3\\right)!}\\left(3x^{2}\\right)^{2}\\left(-7x\\right)^{3}+\\frac{5!}{4!\\left(5-4\\right)!}\\left(3x^{2}\\right)^{1}\\left(-7x\\right)^{4}+\\frac{5!}{5!\\left(5-5\\right)!}\\left(3x^{2}\\right)^{0}\\left(-7x\\right)^{5}"
}
],
"meta": {
"interimType": "Expand Apply Binom 0Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\frac{5!}{0!\\left(5-0\\right)!}\\left(3x^{2}\\right)^{5}\\left(-7x\\right)^{0}:{\\quad}243x^{10}$$",
"input": "\\frac{5!}{0!\\left(5-0\\right)!}\\left(3x^{2}\\right)^{5}\\left(-7x\\right)^{0}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$",
"secondary": [
"$$\\left(-7x\\right)^{0}=1$$"
],
"result": "=1\\cdot\\:\\frac{5!}{0!\\left(5-0\\right)!}\\left(3x^{2}\\right)^{5}"
},
{
"type": "interim",
"title": "$$\\frac{5!}{0!\\left(5-0\\right)!}=1$$",
"input": "\\frac{5!}{0!\\left(5-0\\right)!}",
"steps": [
{
"type": "interim",
"title": "$$0!\\left(5-0\\right)!=5!$$",
"input": "0!\\left(5-0\\right)!",
"steps": [
{
"type": "step",
"primary": "Subtract the numbers: $$5-0=5$$",
"result": "=0!\\cdot\\:5!"
},
{
"type": "step",
"primary": "Apply factorial rule: $$0!=1$$",
"result": "=1\\cdot\\:5!"
},
{
"type": "step",
"primary": "Multiply: $$1\\cdot\\:5!=5!$$",
"result": "=5!"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MwokgvWMKF8XskTEi6KGY80ag8T1MwTer44+aCS/ZFCju+71U4xBfrnsSqWN+7gGwjfOsWcVzDMejC+1+vA7TWcqZUKpIlb03vrjW5aMHcA="
}
},
{
"type": "step",
"result": "=\\frac{5!}{5!}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{a}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PWecMy2etyoRaZbhezcIQHWNBd9fqvrHPwPkioZTO9irju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nQtrKojtd1gDQDAF7vBhgHh8ZX6XNwohDavvG9gq2Jjl"
}
},
{
"type": "step",
"result": "=1\\cdot\\:1\\cdot\\:\\left(3x^{2}\\right)^{5}"
},
{
"type": "interim",
"title": "$$\\left(3x^{2}\\right)^{5}=3^{5}x^{10}$$",
"input": "\\left(3x^{2}\\right)^{5}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=3^{5}\\left(x^{2}\\right)^{5}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\left(x^{2}\\right)^{5}:{\\quad}x^{10}$$",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=x^{2\\cdot\\:5}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:5=10$$",
"result": "=x^{10}"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=3^{5}x^{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bqGxPOEIcIwgoH3RvEWHTVXTSum/z5kLpMzXS1UJIeyxnOX6rc4jxTboLXQIC6p08tfJDZGe9dKzKy+sisZTS+8xJfPuwX1Uw6h0TOogEM1k4iX0/WeGtwCh8jxRTm/CJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "step",
"result": "=3^{5}\\cdot\\:1\\cdot\\:1\\cdot\\:x^{10}"
},
{
"type": "step",
"primary": "Refine",
"result": "=243x^{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\left(-7x\\right)^{1}:{\\quad}-2835x^{9}$$",
"input": "\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\left(-7x\\right)^{1}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$a^{1}=a$$",
"secondary": [
"$$\\left(-7x\\right)^{1}=-7x$$"
],
"result": "=\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\left(-7x\\right)"
},
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\frac{5!}{1!\\left(5-1\\right)!}\\left(3x^{2}\\right)^{4}\\cdot\\:7x"
},
{
"type": "interim",
"title": "$$\\left(3x^{2}\\right)^{4}{\\quad:\\quad}3^{4}x^{8}$$",
"result": "=-\\frac{5!}{1!\\left(5-1\\right)!}\\cdot\\:3^{4}x^{8}\\cdot\\:7x",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(ab\\right)^{c}=a^{c}b^{c}$$",
"secondary": [
"$$\\left(3x^{2}\\right)^{4}=3^{4}\\left(x^{2}\\right)^{4}$$"
],
"result": "=3^{4}\\left(x^{2}\\right)^{4}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"secondary": [
"$$\\left(x^{2}\\right)^{4}=x^{2\\cdot\\:4}=x^{8}$$"
],
"result": "=3^{4}x^{8}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{8}x=\\:x^{8+1}$$"
],
"result": "=-\\frac{5!}{1!\\left(5-1\\right)!}\\cdot\\:3^{4}\\cdot\\:7x^{8+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$8+1=9$$",
"result": "=-\\frac{5!}{1!\\left(5-1\\right)!}\\cdot\\:3^{4}\\cdot\\:7x^{9}"
},
{
"type": "interim",
"title": "$$\\frac{5!}{1!\\left(5-1\\right)!}=\\frac{5}{1!}$$",
"input": "\\frac{5!}{1!\\left(5-1\\right)!}",
"steps": [
{
"type": "step",
"primary": "Subtract the numbers: $$5-1=4$$",
"result": "=\\frac{5!}{1!\\cdot\\:4!}"
},
{
"type": "step",
"primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$",
"secondary": [
"$$\\frac{5!}{4!}=5$$"
],
"result": "=\\frac{5}{1!}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f5Z/C3InBALOMHY6G6Sf+zkRAz87omiy+wxqP64+EMSrju+5Z51e/ZZSD3gRHwjBrhbUhU7Vyjqz/ZGuq885VT/L0MoYg+CUn6oyL3EO7Ypul0tf7Si2wrxKkkcpohEH0VU8Y6XSNTtgRkBxH6mWQYV1O4tzqeY7aVRTflsUZHM="
}
},
{
"type": "step",
"result": "=-3^{4}\\cdot\\:7\\cdot\\:\\frac{5}{1!}x^{9}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=-\\frac{5\\cdot\\:3^{4}\\cdot\\:7x^{9}}{1!}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$5\\cdot\\:7=35$$",
"result": "=-\\frac{3^{4}\\cdot\\:35x^{9}}{1!}"
},
{
"type": "interim",
"title": "$$35\\cdot\\:3^{4}x^{9}=2835x^{9}$$",
"input": "35\\cdot\\:3^{4}x^{9}",
"steps": [
{
"type": "step",
"primary": "$$3^{4}=81$$",
"result": "=35\\cdot\\:81x^{9}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$35\\cdot\\:81=2835$$",
"result": "=2835x^{9}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jxUw54SFLQ1TQwp5JuKU0f1Y37aFHEmEwD6gNYpbliNwkKGJWEPFPk38sdJMsyPIApc1YI3hyypF+3Pj+K4h2hHgslcW6W5sMZRDBRCaP+D84g2FphudG+1XDN9fjpeil0hLmyeYNu96DdjDrgaPqw=="
}
},
{
"type": "step",
"result": "=-\\frac{2835x^{9}}{1!}"
},
{
"type": "step",
"primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$",
"secondary": [
"$$1!=1$$"
],
"result": "=-\\frac{2835x^{9}}{1}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{1}=a$$",
"result": "=-2835x^{9}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\frac{5!}{2!\\left(5-2\\right)!}\\left(3x^{2}\\right)^{3}\\left(-7x\\right)^{2}:{\\quad}13230x^{8}$$",
"input": "\\frac{5!}{2!\\left(5-2\\right)!}\\left(3x^{2}\\right)^{3}\\left(-7x\\right)^{2}",
"steps": [
{
"type": "interim",
"title": "$$\\frac{5!}{2!\\left(5-2\\right)!}=\\frac{20}{2!}$$",
"input": "\\frac{5!}{2!\\left(5-2\\right)!}",
"steps": [
{
"type": "step",
"primary": "Subtract the numbers: $$5-2=3$$",
"result": "=\\frac{5!}{2!\\cdot\\:3!}"
},
{
"type": "step",
"primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$",
"secondary": [
"$$\\frac{5!}{3!}=5\\cdot\\:4$$"
],
"result": "=\\frac{5\\cdot\\:4}{2!}"
},
{
"type": "step",
"primary": "Refine",
"result": "=\\frac{20}{2!}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n3FmoZ6eADfOpmZSwpdYK+gDk5cEadJ9VlqSrg22yQyrju+5Z51e/ZZSD3gRHwjBeiuF3n9ax1ott9J/0tUf8/8//6/nV5O4fb8Xgwi7maqPDgDZhikHx/w+ZSw3R9RcXHp0eMnhEqp8VFcQIjhvhHh2ZKnWpxIymVLpM1P79gg="
}
},
{
"type": "step",
"result": "=\\frac{20}{2!}\\left(3x^{2}\\right)^{3}\\left(-7x\\right)^{2}"
},
{
"type": "interim",
"title": "$$\\left(3x^{2}\\right)^{3}=3^{3}x^{6}$$",
"input": "\\left(3x^{2}\\right)^{3}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=3^{3}\\left(x^{2}\\right)^{3}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\left(x^{2}\\right)^{3}:{\\quad}x^{6}$$",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=x^{2\\cdot\\:3}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:3=6$$",
"result": "=x^{6}"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=3^{3}x^{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZRkoRnfk0jphmyffn03aK1XTSum/z5kLpMzXS1UJIexB37Gv+QVnUvB0vpeRXo49/SrXi7VL6w/e9DH0I82kqkdDdd7BdFLIMjLYiCeYWxywyedKQ2BtLYGL0M9A8GoK"
}
},
{
"type": "step",
"result": "=3^{3}\\cdot\\:\\frac{20}{2!}x^{6}\\left(-7x\\right)^{2}"
},
{
"type": "interim",
"title": "$$\\left(-7x\\right)^{2}=7^{2}x^{2}$$",
"input": "\\left(-7x\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even",
"secondary": [
"$$\\left(-7x\\right)^{2}=\\left(7x\\right)^{2}$$"
],
"result": "=\\left(7x\\right)^{2}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=7^{2}x^{2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78YvuSBeed9/oNLmcjzhy5N13jtrSFDx+UNsawjlOjV1JZZKcGUWD2lJswqgxGqt0Ec7ShOedm97LMngC0LVkYwTZ7eXvkkAoDD94ITCftasuGhvdsDe+55cUO+VX2gUd"
}
},
{
"type": "step",
"result": "=3^{3}\\cdot\\:7^{2}\\cdot\\:\\frac{20}{2!}x^{6}x^{2}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{6}x^{2}=\\:x^{6+2}$$"
],
"result": "=\\frac{20}{2!}\\cdot\\:3^{3}\\cdot\\:7^{2}x^{6+2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$6+2=8$$",
"result": "=\\frac{20}{2!}\\cdot\\:3^{3}\\cdot\\:7^{2}x^{8}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{20\\cdot\\:3^{3}\\cdot\\:7^{2}x^{8}}{2!}"
},
{
"type": "interim",
"title": "$$20\\cdot\\:3^{3}\\cdot\\:7^{2}x^{8}=26460x^{8}$$",
"input": "20\\cdot\\:3^{3}\\cdot\\:7^{2}x^{8}",
"steps": [
{
"type": "step",
"primary": "$$3^{3}=27$$",
"result": "=7^{2}\\cdot\\:20\\cdot\\:27x^{8}"
},
{
"type": "step",
"primary": "$$7^{2}=49$$",
"result": "=20\\cdot\\:27\\cdot\\:49x^{8}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$20\\cdot\\:27\\cdot\\:49=26460$$",
"result": "=26460x^{8}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75EgtsuadzPXP7ZRp0A5LCjq8F08IDhe32tQwdtgVg498kR7hsO/rTOTBE0w4+r1RZOpjW2EvHXPq6tKyfIu0FtcZb0OnwUGfLplAZu/dhEqF3uywltbe0TxAoX7T04/eLnHgFbE9/1QcMihurcuaRiyLdrbaBHX3P2h8cfOgQCjBRsgJ27t82dl54+j2dJQW"
}
},
{
"type": "step",
"result": "=\\frac{26460x^{8}}{2!}"
},
{
"type": "interim",
"title": "$$2!=2$$",
"input": "2!",
"steps": [
{
"type": "step",
"primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$",
"secondary": [
"$$2!=1\\cdot\\:2$$"
],
"result": "=1\\cdot\\:2"
},
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:2=2$$",
"result": "=2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7v6/eVh8vKMXnTnxuN3zFTMzBWJotReR4P4m6RE6FZ2MEnP7Xv4jqQdsgR6rCHLiZRJ9TLnjGcV7xuLm79Wde5Q=="
}
},
{
"type": "step",
"result": "=\\frac{26460x^{8}}{2}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{26460}{2}=13230$$",
"result": "=13230x^{8}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\frac{5!}{3!\\left(5-3\\right)!}\\left(3x^{2}\\right)^{2}\\left(-7x\\right)^{3}:{\\quad}-30870x^{7}$$",
"input": "\\frac{5!}{3!\\left(5-3\\right)!}\\left(3x^{2}\\right)^{2}\\left(-7x\\right)^{3}",
"steps": [
{
"type": "interim",
"title": "$$\\frac{5!}{3!\\left(5-3\\right)!}=\\frac{20}{2!}$$",
"input": "\\frac{5!}{3!\\left(5-3\\right)!}",
"steps": [
{
"type": "step",
"primary": "Subtract the numbers: $$5-3=2$$",
"result": "=\\frac{5!}{3!\\cdot\\:2!}"
},
{
"type": "step",
"primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$",
"secondary": [
"$$\\frac{5!}{3!}=5\\cdot\\:4$$"
],
"result": "=\\frac{5\\cdot\\:4}{2!}"
},
{
"type": "step",
"primary": "Refine",
"result": "=\\frac{20}{2!}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OlFMhK3TIaJe5Gd6NetMWaopdjM+QCA2DLIQekqv9Hqrju+5Z51e/ZZSD3gRHwjBeiuF3n9ax1ott9J/0tUf8/8//6/nV5O4fb8Xgwi7mapwVofzIItVnps+BOXRavpqmvLrq9m2T2oGPK0vGoghhnh2ZKnWpxIymVLpM1P79gg="
}
},
{
"type": "step",
"result": "=\\frac{20}{2!}\\left(3x^{2}\\right)^{2}\\left(-7x\\right)^{3}"
},
{
"type": "interim",
"title": "$$\\left(3x^{2}\\right)^{2}=3^{2}x^{4}$$",
"input": "\\left(3x^{2}\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=3^{2}\\left(x^{2}\\right)^{2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\left(x^{2}\\right)^{2}:{\\quad}x^{4}$$",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=x^{2\\cdot\\:2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:2=4$$",
"result": "=x^{4}"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=3^{2}x^{4}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XXedBRdf1W3pIn8dcJrBgFXTSum/z5kLpMzXS1UJIex6aza7XLrlvYDV0tAJrhr7A2MIkDC/Nt4Kj11AaxXgLP7orB3Zf9tH6c3YjlAplhhdQsRSOyhcxCIJGV3Igiyq"
}
},
{
"type": "step",
"result": "=3^{2}\\cdot\\:\\frac{20}{2!}x^{4}\\left(-7x\\right)^{3}"
},
{
"type": "interim",
"title": "$$\\left(-7x\\right)^{3}=-7^{3}x^{3}$$",
"input": "\\left(-7x\\right)^{3}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd",
"secondary": [
"$$\\left(-7x\\right)^{3}=-\\left(7x\\right)^{3}$$"
],
"result": "=-\\left(7x\\right)^{3}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"secondary": [
"$$\\left(7x\\right)^{3}=7^{3}x^{3}$$"
],
"result": "=-7^{3}x^{3}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xMEaBU+RbkOqaqo/Xn4oEN13jtrSFDx+UNsawjlOjV383QBZMNPbYkun7dBkxcj/ZhicctV/NaBeEeor5HUYWiiLYKphqaBbzFY+hWKUn8HD32FWonxGTvOcH2P7Y3uE"
}
},
{
"type": "step",
"result": "=3^{2}\\cdot\\:\\frac{20}{2!}x^{4}\\left(-7^{3}x^{3}\\right)"
},
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-\\frac{20}{2!}\\cdot\\:3^{2}x^{4}\\cdot\\:7^{3}x^{3}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{4}x^{3}=\\:x^{4+3}$$"
],
"result": "=-\\frac{20}{2!}\\cdot\\:3^{2}\\cdot\\:7^{3}x^{4+3}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$4+3=7$$",
"result": "=-\\frac{20}{2!}\\cdot\\:3^{2}\\cdot\\:7^{3}x^{7}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=-\\frac{20\\cdot\\:3^{2}\\cdot\\:7^{3}x^{7}}{2!}"
},
{
"type": "interim",
"title": "$$20\\cdot\\:3^{2}\\cdot\\:7^{3}x^{7}=61740x^{7}$$",
"input": "20\\cdot\\:3^{2}\\cdot\\:7^{3}x^{7}",
"steps": [
{
"type": "step",
"primary": "$$3^{2}=9$$",
"result": "=7^{3}\\cdot\\:20\\cdot\\:9x^{7}"
},
{
"type": "step",
"primary": "$$7^{3}=343$$",
"result": "=20\\cdot\\:9\\cdot\\:343x^{7}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$20\\cdot\\:9\\cdot\\:343=61740$$",
"result": "=61740x^{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NpHZHAJwYWhMeM+XFYfsTTtGz1R+vhot3wSmAGxhHIV8kR7hsO/rTOTBE0w4+r1RUWi/oqxS2IKYY446wHsdvhd90fhhwP9VzfIIwFbn9ouF3uywltbe0TxAoX7T04/eucx87Md/Fyv4TLaagsI7WL87LarQ5l+Netyg2tXLuCfQjUcXS9wUNR7xIXvV0A2z"
}
},
{
"type": "step",
"result": "=-\\frac{61740x^{7}}{2!}"
},
{
"type": "interim",
"title": "$$2!=2$$",
"input": "2!",
"steps": [
{
"type": "step",
"primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$",
"secondary": [
"$$2!=1\\cdot\\:2$$"
],
"result": "=1\\cdot\\:2"
},
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:2=2$$",
"result": "=2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7v6/eVh8vKMXnTnxuN3zFTMzBWJotReR4P4m6RE6FZ2MEnP7Xv4jqQdsgR6rCHLiZRJ9TLnjGcV7xuLm79Wde5Q=="
}
},
{
"type": "step",
"result": "=-\\frac{61740x^{7}}{2}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{61740}{2}=30870$$",
"result": "=-30870x^{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\frac{5!}{4!\\left(5-4\\right)!}\\left(3x^{2}\\right)^{1}\\left(-7x\\right)^{4}:{\\quad}36015x^{6}$$",
"input": "\\frac{5!}{4!\\left(5-4\\right)!}\\left(3x^{2}\\right)^{1}\\left(-7x\\right)^{4}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$a^{1}=a$$",
"secondary": [
"$$\\left(3x^{2}\\right)^{1}=3x^{2}$$"
],
"result": "=3\\cdot\\:\\frac{5!}{4!\\left(5-4\\right)!}x^{2}\\left(-7x\\right)^{4}"
},
{
"type": "interim",
"title": "$$\\frac{5!}{4!\\left(5-4\\right)!}=\\frac{5}{1!}$$",
"input": "\\frac{5!}{4!\\left(5-4\\right)!}",
"steps": [
{
"type": "step",
"primary": "Subtract the numbers: $$5-4=1$$",
"result": "=\\frac{5!}{4!\\cdot\\:1!}"
},
{
"type": "step",
"primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$",
"secondary": [
"$$\\frac{5!}{4!}=5$$"
],
"result": "=\\frac{5}{1!}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s736V5IQ4wCOzUO8oLf5R0/U4QonqtsWqeLv4Cgbcmdxqrju+5Z51e/ZZSD3gRHwjBrhbUhU7Vyjqz/ZGuq885VT/L0MoYg+CUn6oyL3EO7YrddiYtvGnqV01DCxRAcscmuStfOZX8fYrUai2Dz5F3oIV1O4tzqeY7aVRTflsUZHM="
}
},
{
"type": "step",
"result": "=3\\cdot\\:\\frac{5}{1!}x^{2}\\left(-7x\\right)^{4}"
},
{
"type": "interim",
"title": "$$\\left(-7x\\right)^{4}=7^{4}x^{4}$$",
"input": "\\left(-7x\\right)^{4}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even",
"secondary": [
"$$\\left(-7x\\right)^{4}=\\left(7x\\right)^{4}$$"
],
"result": "=\\left(7x\\right)^{4}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=7^{4}x^{4}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vYdUwyCCccGSjHkJ5x2T8N13jtrSFDx+UNsawjlOjV0MRbikmhUlmRBHQ2AGrdVpcTZnm/8YWra/1y/CNJ8TTMyajDt4irSC/BTc2EmigJswt+5grAHqXsUcSNMjgSB0"
}
},
{
"type": "step",
"result": "=7^{4}\\cdot\\:3\\cdot\\:\\frac{5}{1!}x^{4}x^{2}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{2}x^{4}=\\:x^{2+4}$$"
],
"result": "=\\frac{5}{1!}\\cdot\\:3\\cdot\\:7^{4}x^{2+4}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$2+4=6$$",
"result": "=\\frac{5}{1!}\\cdot\\:3\\cdot\\:7^{4}x^{6}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{5\\cdot\\:3\\cdot\\:7^{4}x^{6}}{1!}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$5\\cdot\\:3=15$$",
"result": "=\\frac{7^{4}\\cdot\\:15x^{6}}{1!}"
},
{
"type": "interim",
"title": "$$15\\cdot\\:7^{4}x^{6}=36015x^{6}$$",
"input": "15\\cdot\\:7^{4}x^{6}",
"steps": [
{
"type": "step",
"primary": "$$7^{4}=2401$$",
"result": "=15\\cdot\\:2401x^{6}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$15\\cdot\\:2401=36015$$",
"result": "=36015x^{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QmfoWI8DgcK/eHFDSVBsyq/Uaaceamqua44+OzGk72hwkKGJWEPFPk38sdJMsyPI+9eNKdP6sh1kZk+RuqtFwaUAqfZOfMCaHxUGRIjWXxMTQq4H3v16ueOQPWQejQRIo9IcqJ7b2ZnHSYg3Vnp7Vg=="
}
},
{
"type": "step",
"result": "=\\frac{36015x^{6}}{1!}"
},
{
"type": "step",
"primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$",
"secondary": [
"$$1!=1$$"
],
"result": "=\\frac{36015x^{6}}{1}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{1}=a$$",
"result": "=36015x^{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "interim",
"title": "Simplify $$\\frac{5!}{5!\\left(5-5\\right)!}\\left(3x^{2}\\right)^{0}\\left(-7x\\right)^{5}:{\\quad}-16807x^{5}$$",
"input": "\\frac{5!}{5!\\left(5-5\\right)!}\\left(3x^{2}\\right)^{0}\\left(-7x\\right)^{5}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$",
"secondary": [
"$$\\left(3x^{2}\\right)^{0}=1$$"
],
"result": "=1\\cdot\\:\\frac{5!}{5!\\left(5-5\\right)!}\\left(-7x\\right)^{5}"
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=1\\cdot\\:\\frac{5!\\left(-7x\\right)^{5}}{5!\\left(5-5\\right)!}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$5!$$",
"result": "=1\\cdot\\:\\frac{\\left(-7x\\right)^{5}}{\\left(5-5\\right)!}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{\\left(-7x\\right)^{5}}{\\left(5-5\\right)!}:{\\quad}-16807x^{5}$$",
"input": "\\frac{\\left(-7x\\right)^{5}}{\\left(5-5\\right)!}",
"steps": [
{
"type": "interim",
"title": "$$\\left(-7x\\right)^{5}=-\\left(7x\\right)^{5}$$",
"input": "\\left(-7x\\right)^{5}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd",
"secondary": [
"$$\\left(-7x\\right)^{5}=-\\left(7x\\right)^{5}$$"
],
"result": "=-\\left(7x\\right)^{5}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zFR9q+8Dk0Ggojtw/ixwdt13jtrSFDx+UNsawjlOjV1gL+D9rqwk59qWhBECiSxH/z//r+dXk7h9vxeDCLuZqmfjfjE38QdPVh2P8+3WNtPoYTi3bKSyy26aYJz3Y9y4"
}
},
{
"type": "step",
"result": "=\\frac{-\\left(7x\\right)^{5}}{\\left(5-5\\right)!}"
},
{
"type": "interim",
"title": "$$\\left(5-5\\right)!=1$$",
"input": "\\left(5-5\\right)!",
"steps": [
{
"type": "step",
"primary": "Subtract the numbers: $$5-5=0$$",
"result": "=0!"
},
{
"type": "step",
"primary": "Apply factorial rule: $$0!=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UYqdMQX/bBuG52oF2WEcnHWD310L1+P2yDQQfMEhENH7xkO7yIXgxJ7C3jKazWUhBSyaJWaCEhDhvCCW1Y4x5iS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=\\frac{-\\left(7x\\right)^{5}}{1}"
},
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$",
"result": "=-\\frac{\\left(7x\\right)^{5}}{1}"
},
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{1}=a$$",
"result": "=-\\left(7x\\right)^{5}"
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=-7^{5}x^{5}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "$$7^{5}=16807$$",
"result": "=-16807x^{5}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=1\\cdot\\:\\left(-16807x^{5}\\right)"
},
{
"type": "step",
"primary": "Refine",
"result": "=-16807x^{5}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=243x^{10}-2835x^{9}+13230x^{8}-30870x^{7}+36015x^{6}-16807x^{5}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UxyIjAYLNsR4LNrU/XDEpHyRHuGw7+tM5METTDj6vVE3CXpyb+Iu3cMrCn545PU89IkjFtSQVMCEel3Dz3uPAQr9xiZshrcjb6UpnQDwUFdYHIYSXiAXxp8eK3vCBePYma6s9rQGAM7a5c+QuBTCsM0AoIL8rSt1wm3N89qpYbB6j4AEaA+klFXb6y9pRrBsm0MJdmCkjl/r8viyegzs5qyB/oa7JjSrSqMAjbE5uIW6if1VkxFFkPEpDCjzhUcS0hkKtt6x/fvOrlVnTlR7uQ=="
}
},
{
"type": "step",
"result": "=\\left(243x^{10}-2835x^{9}+13230x^{8}-30870x^{7}+36015x^{6}-16807x^{5}\\right)\\left(6x-7\\right)"
},
{
"type": "step",
"primary": "Distribute parentheses",
"result": "=243x^{10}\\cdot\\:6x+243x^{10}\\left(-7\\right)+\\left(-2835x^{9}\\right)\\cdot\\:6x+\\left(-2835x^{9}\\right)\\left(-7\\right)+13230x^{8}\\cdot\\:6x+13230x^{8}\\left(-7\\right)+\\left(-30870x^{7}\\right)\\cdot\\:6x+\\left(-30870x^{7}\\right)\\left(-7\\right)+36015x^{6}\\cdot\\:6x+36015x^{6}\\left(-7\\right)+\\left(-16807x^{5}\\right)\\cdot\\:6x+\\left(-16807x^{5}\\right)\\left(-7\\right)",
"meta": {
"title": {
"extension": "Multiply each of the terms within the first parentheses<br/>by each of the terms within the second parentheses left to right"
}
}
},
{
"type": "step",
"primary": "Apply minus-plus rules",
"secondary": [
"$$+\\left(-a\\right)=-a,\\:\\:\\left(-a\\right)\\left(-b\\right)=ab$$"
],
"result": "=243\\cdot\\:6x^{10}x-243\\cdot\\:7x^{10}-2835\\cdot\\:6x^{9}x+2835\\cdot\\:7x^{9}+13230\\cdot\\:6x^{8}x-13230\\cdot\\:7x^{8}-30870\\cdot\\:6x^{7}x+30870\\cdot\\:7x^{7}+36015\\cdot\\:6x^{6}x-36015\\cdot\\:7x^{6}-16807\\cdot\\:6x^{5}x+16807\\cdot\\:7x^{5}"
},
{
"type": "interim",
"title": "Simplify $$243\\cdot\\:6x^{10}x-243\\cdot\\:7x^{10}-2835\\cdot\\:6x^{9}x+2835\\cdot\\:7x^{9}+13230\\cdot\\:6x^{8}x-13230\\cdot\\:7x^{8}-30870\\cdot\\:6x^{7}x+30870\\cdot\\:7x^{7}+36015\\cdot\\:6x^{6}x-36015\\cdot\\:7x^{6}-16807\\cdot\\:6x^{5}x+16807\\cdot\\:7x^{5}:{\\quad}1458x^{11}-18711x^{10}+99225x^{9}-277830x^{8}+432180x^{7}-352947x^{6}+117649x^{5}$$",
"input": "243\\cdot\\:6x^{10}x-243\\cdot\\:7x^{10}-2835\\cdot\\:6x^{9}x+2835\\cdot\\:7x^{9}+13230\\cdot\\:6x^{8}x-13230\\cdot\\:7x^{8}-30870\\cdot\\:6x^{7}x+30870\\cdot\\:7x^{7}+36015\\cdot\\:6x^{6}x-36015\\cdot\\:7x^{6}-16807\\cdot\\:6x^{5}x+16807\\cdot\\:7x^{5}",
"result": "=1458x^{11}-18711x^{10}+99225x^{9}-277830x^{8}+432180x^{7}-352947x^{6}+117649x^{5}",
"steps": [
{
"type": "interim",
"title": "$$243\\cdot\\:6x^{10}x=1458x^{11}$$",
"input": "243\\cdot\\:6x^{10}x",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$243\\cdot\\:6=1458$$",
"result": "=1458x^{10}x"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{10}x=\\:x^{10+1}$$"
],
"result": "=1458x^{10+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$10+1=11$$",
"result": "=1458x^{11}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7K8dh6yU+chPKkPIeNOxrRoQQcs4+B3dxMD0+SQ65e1bMwViaLUXkeD+JukROhWdjmMfDIf80ghjilLGZebn7/xxZL0IQQPbbp13tAqSBDcdR1ZC1gKefnu+wv6j1MC+6Y0bmeHUdlE6kadSAYsBxfA=="
}
},
{
"type": "interim",
"title": "$$243\\cdot\\:7x^{10}=1701x^{10}$$",
"input": "243\\cdot\\:7x^{10}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$243\\cdot\\:7=1701$$",
"result": "=1701x^{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7j+Me5FkMXoqNb+MQaAY8xuP78sN6s+TxU1X701cSbd+jkVi15I8rBefLi4Iyt2wrj+KE/FJDtm0H9zOkBHk/j1oTQB1vlgt9wc3NSrXVOdi3nM8FpjQCGGlLbxQ9yVzdDR6ultXbiYwQzcqgy36cCg=="
}
},
{
"type": "interim",
"title": "$$2835\\cdot\\:6x^{9}x=17010x^{10}$$",
"input": "2835\\cdot\\:6x^{9}x",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$2835\\cdot\\:6=17010$$",
"result": "=17010x^{9}x"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{9}x=\\:x^{9+1}$$"
],
"result": "=17010x^{9+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$9+1=10$$",
"result": "=17010x^{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aXnMs/uUSOXY8vJzU00wGxC29AYplC36+p+AWps/+7XMwViaLUXkeD+JukROhWdjbKESlDhToV/19PMY4JVS2btewqyNvf0kHo0+VJubWp8ZMUkZjQlYSNwzP7Sz54qQgo+pcRT3xQyGNM3NQm6OaA=="
}
},
{
"type": "interim",
"title": "$$2835\\cdot\\:7x^{9}=19845x^{9}$$",
"input": "2835\\cdot\\:7x^{9}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$2835\\cdot\\:7=19845$$",
"result": "=19845x^{9}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RHep8BpzU4DxUbaDLL/fNpd250f0GBn8Qd1oVcl//l+jkVi15I8rBefLi4Iyt2wrHaKh3hSIGqU1pTqy2yWawpZj6JvXaCaOenLctifLaeKZcma07/qM7clcm545f5q0kzBVQoRX0K+OV7hLTzhxYA=="
}
},
{
"type": "interim",
"title": "$$13230\\cdot\\:6x^{8}x=79380x^{9}$$",
"input": "13230\\cdot\\:6x^{8}x",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$13230\\cdot\\:6=79380$$",
"result": "=79380x^{8}x"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{8}x=\\:x^{8+1}$$"
],
"result": "=79380x^{8+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$8+1=9$$",
"result": "=79380x^{9}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QMgFtOLHlKf+vg3HQHLwOfOCK85joz8iBw85tXy1MUpwkKGJWEPFPk38sdJMsyPINW1/hwMOH7TC153qlXchhoAyRKusrHHcX+hbezCgiU6/LjWLJRSloe+3SCoWTwpXQN+8QTGxa8H5sYVlGl/nWA=="
}
},
{
"type": "interim",
"title": "$$13230\\cdot\\:7x^{8}=92610x^{8}$$",
"input": "13230\\cdot\\:7x^{8}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$13230\\cdot\\:7=92610$$",
"result": "=92610x^{8}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l698cnYRX28AGgvmY85L4ZZZLbZrgaMHW+gEI5N0HMzMwViaLUXkeD+JukROhWdjtMrGQFuzgZ/PcKQOAORGWPZutJENm7Be9tkn2YRFbVim4pfBWIEmn9Iy44JYIGb2U7HQK7T/Y+f3YYmwReKGIg=="
}
},
{
"type": "interim",
"title": "$$30870\\cdot\\:6x^{7}x=185220x^{8}$$",
"input": "30870\\cdot\\:6x^{7}x",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$30870\\cdot\\:6=185220$$",
"result": "=185220x^{7}x"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{7}x=\\:x^{7+1}$$"
],
"result": "=185220x^{7+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$7+1=8$$",
"result": "=185220x^{8}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7t4hw11DhI1S9cYpVeVLv39200QHZl609cqvMvHEoEv5wkKGJWEPFPk38sdJMsyPIgvUD1W93OmKAeeBCF6gYTSulMzZahlFUlY0r5cqKo0u/inXG/V2BSWIEJIxQEwpUOBqglKmJ5TOxsqWXPlYPiA=="
}
},
{
"type": "interim",
"title": "$$30870\\cdot\\:7x^{7}=216090x^{7}$$",
"input": "30870\\cdot\\:7x^{7}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$30870\\cdot\\:7=216090$$",
"result": "=216090x^{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XXMS9dpEQoz3yYWtWGQ25lw6FQTjqOFY2dpMCUby1qnMwViaLUXkeD+JukROhWdjNhvcV/Ak2CkWAN12ZMdJRUYcry/xRyAB9kQVpvW7dLu0V4MPTSQkeQX9iEpWxuczYslubWOAFzEBXTTPo6kVpw=="
}
},
{
"type": "interim",
"title": "$$36015\\cdot\\:6x^{6}x=216090x^{7}$$",
"input": "36015\\cdot\\:6x^{6}x",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$36015\\cdot\\:6=216090$$",
"result": "=216090x^{6}x"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{6}x=\\:x^{6+1}$$"
],
"result": "=216090x^{6+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$6+1=7$$",
"result": "=216090x^{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ooMCK6rMlAzw5ZBdUyRdzi/ux2GgTq1pq3yirp1uJX9wkKGJWEPFPk38sdJMsyPIJPj0KOZRif4vW1BVCLOfWQ1hsRYS4ijAARd5keMDMcJM+ly89+gLaPllGO/WNmn0kNmgboI9GiFtDxy+NzxQkg=="
}
},
{
"type": "interim",
"title": "$$36015\\cdot\\:7x^{6}=252105x^{6}$$",
"input": "36015\\cdot\\:7x^{6}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$36015\\cdot\\:7=252105$$",
"result": "=252105x^{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C6JQxOnWwCVwzB8OjzcaHfIsZ5hHF6bZxqb3cGR9OSHMwViaLUXkeD+JukROhWdjsiTb3KegEvMqbvuRy9POqcp8n/A/IU8nFU4MC3tf1Clarfc/eUAXv0pV0xW8LQEiwCNdEao9e7afKVwjCNanrg=="
}
},
{
"type": "interim",
"title": "$$16807\\cdot\\:6x^{5}x=100842x^{6}$$",
"input": "16807\\cdot\\:6x^{5}x",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$16807\\cdot\\:6=100842$$",
"result": "=100842x^{5}x"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$x^{5}x=\\:x^{5+1}$$"
],
"result": "=100842x^{5+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$5+1=6$$",
"result": "=100842x^{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g/Oq3wSOZj2c6Gjt0UohNZKu1LXKNlyF1Lk0Plo/ifFwkKGJWEPFPk38sdJMsyPIZdYgadSpJYhVCd7HeysXvKAbfWhrdt+2Zm3wF1iUEIXfZNELy85/8AUF/NrojEFSJlhYdl/m5k1rq9al8kYNrw=="
}
},
{
"type": "interim",
"title": "$$16807\\cdot\\:7x^{5}=117649x^{5}$$",
"input": "16807\\cdot\\:7x^{5}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$16807\\cdot\\:7=117649$$",
"result": "=117649x^{5}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zVdgcFUCeIQPBTrVZDLVZqT0+OO7WzeBmYKPy9hlv6HMwViaLUXkeD+JukROhWdj7/grXvFJeK6IFlK7hBE7vYKzoXF3t7R4g4xcuAzN7s0mQbXVhgUYgnKY4o41ejcceksrLQ3YVuuK0PIx0rEcnA=="
}
},
{
"type": "step",
"result": "=1458x^{11}-1701x^{10}-17010x^{10}+19845x^{9}+79380x^{9}-92610x^{8}-185220x^{8}+216090x^{7}+216090x^{7}-252105x^{6}-100842x^{6}+117649x^{5}"
},
{
"type": "step",
"primary": "Add similar elements: $$-1701x^{10}-17010x^{10}=-18711x^{10}$$",
"result": "=1458x^{11}-18711x^{10}+19845x^{9}+79380x^{9}-92610x^{8}-185220x^{8}+216090x^{7}+216090x^{7}-252105x^{6}-100842x^{6}+117649x^{5}"
},
{
"type": "step",
"primary": "Add similar elements: $$-252105x^{6}-100842x^{6}=-352947x^{6}$$",
"result": "=1458x^{11}-18711x^{10}+19845x^{9}+79380x^{9}-92610x^{8}-185220x^{8}+216090x^{7}+216090x^{7}-352947x^{6}+117649x^{5}"
},
{
"type": "step",
"primary": "Add similar elements: $$216090x^{7}+216090x^{7}=432180x^{7}$$",
"result": "=1458x^{11}-18711x^{10}+19845x^{9}+79380x^{9}-92610x^{8}-185220x^{8}+432180x^{7}-352947x^{6}+117649x^{5}"
},
{
"type": "step",
"primary": "Add similar elements: $$-92610x^{8}-185220x^{8}=-277830x^{8}$$",
"result": "=1458x^{11}-18711x^{10}+19845x^{9}+79380x^{9}-277830x^{8}+432180x^{7}-352947x^{6}+117649x^{5}"
},
{
"type": "step",
"primary": "Add similar elements: $$19845x^{9}+79380x^{9}=99225x^{9}$$",
"result": "=1458x^{11}-18711x^{10}+99225x^{9}-277830x^{8}+432180x^{7}-352947x^{6}+117649x^{5}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Expand Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UxyIjAYLNsR4LNrU/XDEpCAktmB/rNG4enBCoryjsqsJQJZuTAY5js+oqjdT8kslNG9hZy6ca3xPFIcVhc8/iuKhrN6p3sP7VHWAxzrpZIho1KA2TiYXEmTo1g0ULQkvFiNVm4q368hCNtqgmKEZmATPue/ruodGbHk5D/NxfJP6Ir+LgBxiLyiJGbLwViiOEnJ/zNrC2ARoMQpaswsTNqUuQdorwl6H7Avwbqs/L3DhcjiCdaiXOX4AV6d+Q84K"
}
},
{
"type": "step",
"result": "=\\int\\:1458x^{11}-18711x^{10}+99225x^{9}-277830x^{8}+432180x^{7}-352947x^{6}+117649x^{5}dx"
},
{
"type": "step",
"primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$",
"result": "=\\int\\:1458x^{11}dx-\\int\\:18711x^{10}dx+\\int\\:99225x^{9}dx-\\int\\:277830x^{8}dx+\\int\\:432180x^{7}dx-\\int\\:352947x^{6}dx+\\int\\:117649x^{5}dx"
},
{
"type": "interim",
"title": "$$\\int\\:1458x^{11}dx=\\frac{243x^{12}}{2}$$",
"input": "\\int\\:1458x^{11}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=1458\\cdot\\:\\int\\:x^{11}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{11}dx",
"result": "=1458\\cdot\\:\\frac{x^{12}}{12}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{11+1}}{11+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{11+1}}{11+1}:{\\quad}\\frac{x^{12}}{12}$$",
"input": "\\frac{x^{11+1}}{11+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$11+1=12$$",
"result": "=\\frac{x^{12}}{12}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{12}}{12}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71dEic4CRT9bsI3TZ4GZmDFncagboqjyMM+Hi/KaGzT65YlCxobrbz5ibTAG5yuyuO/75tKmtPwNEuMNDNr5dFW7u3QG/FZvVChCbaVZr3Aho3oe/oyhMy2+1TQhDBd2fzGrMGuaCGoCkBTn65ypufg7MKqJIkyqwzoZknRYPhBw"
}
},
{
"type": "interim",
"title": "Simplify $$1458\\cdot\\:\\frac{x^{12}}{12}:{\\quad}\\frac{243x^{12}}{2}$$",
"input": "1458\\cdot\\:\\frac{x^{12}}{12}",
"result": "=\\frac{243x^{12}}{2}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{x^{12}\\cdot\\:1458}{12}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$6$$",
"result": "=\\frac{243x^{12}}{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oAamZRxxmNc91I6q8SA7OXg44WlmCJNpfbocqtAJAAQtOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOLjOke+ckG/yAXhyWPC6cB4/y9DKGIPglJ+qMi9xDu2KaRI7GCp0HQz+zDw23axddEgYUlGfQoRM6ZHKNtlm6vc5ItqquDUase7MrOjkn9JjsIjaxJ4DvjTb2fbKjbvtlQ=="
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "interim",
"title": "$$\\int\\:18711x^{10}dx=1701x^{11}$$",
"input": "\\int\\:18711x^{10}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=18711\\cdot\\:\\int\\:x^{10}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{10}dx",
"result": "=18711\\cdot\\:\\frac{x^{11}}{11}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{10+1}}{10+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{10+1}}{10+1}:{\\quad}\\frac{x^{11}}{11}$$",
"input": "\\frac{x^{10+1}}{10+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$10+1=11$$",
"result": "=\\frac{x^{11}}{11}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{11}}{11}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77hJxFNIlz6KjamiDubtXJtncagboqjyMM+Hi/KaGzT65YlCxobrbz5ibTAG5yuyuO/75tKmtPwNEuMNDNr5dFVBh/6IJkL4yDouXfICYR4Ro3oe/oyhMy2+1TQhDBd2fzGrMGuaCGoCkBTn65ypufg7MKqJIkyqwzoZknRYPhBw"
}
},
{
"type": "interim",
"title": "Simplify $$18711\\cdot\\:\\frac{x^{11}}{11}:{\\quad}1701x^{11}$$",
"input": "18711\\cdot\\:\\frac{x^{11}}{11}",
"result": "=1701x^{11}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{x^{11}\\cdot\\:18711}{11}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{18711}{11}=1701$$",
"result": "=1701x^{11}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
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}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
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},
{
"type": "interim",
"title": "$$\\int\\:99225x^{9}dx=\\frac{19845x^{10}}{2}$$",
"input": "\\int\\:99225x^{9}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=99225\\cdot\\:\\int\\:x^{9}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{9}dx",
"result": "=99225\\cdot\\:\\frac{x^{10}}{10}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{9+1}}{9+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{9+1}}{9+1}:{\\quad}\\frac{x^{10}}{10}$$",
"input": "\\frac{x^{9+1}}{9+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$9+1=10$$",
"result": "=\\frac{x^{10}}{10}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{10}}{10}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
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},
{
"type": "interim",
"title": "Simplify $$99225\\cdot\\:\\frac{x^{10}}{10}:{\\quad}\\frac{19845x^{10}}{2}$$",
"input": "99225\\cdot\\:\\frac{x^{10}}{10}",
"result": "=\\frac{19845x^{10}}{2}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{x^{10}\\cdot\\:99225}{10}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$5$$",
"result": "=\\frac{19845x^{10}}{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
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],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
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},
{
"type": "interim",
"title": "$$\\int\\:277830x^{8}dx=30870x^{9}$$",
"input": "\\int\\:277830x^{8}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=277830\\cdot\\:\\int\\:x^{8}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{8}dx",
"result": "=277830\\cdot\\:\\frac{x^{9}}{9}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{8+1}}{8+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{8+1}}{8+1}:{\\quad}\\frac{x^{9}}{9}$$",
"input": "\\frac{x^{8+1}}{8+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$8+1=9$$",
"result": "=\\frac{x^{9}}{9}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{9}}{9}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
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}
},
{
"type": "interim",
"title": "Simplify $$277830\\cdot\\:\\frac{x^{9}}{9}:{\\quad}30870x^{9}$$",
"input": "277830\\cdot\\:\\frac{x^{9}}{9}",
"result": "=30870x^{9}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{x^{9}\\cdot\\:277830}{9}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{277830}{9}=30870$$",
"result": "=30870x^{9}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
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}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "interim",
"title": "$$\\int\\:432180x^{7}dx=\\frac{108045x^{8}}{2}$$",
"input": "\\int\\:432180x^{7}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=432180\\cdot\\:\\int\\:x^{7}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{7}dx",
"result": "=432180\\cdot\\:\\frac{x^{8}}{8}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{7+1}}{7+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{7+1}}{7+1}:{\\quad}\\frac{x^{8}}{8}$$",
"input": "\\frac{x^{7+1}}{7+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$7+1=8$$",
"result": "=\\frac{x^{8}}{8}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{8}}{8}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s78/zM1Ulm/GZYO5rUZkTS+qo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7offE7fULzLKTed+7Kn4hc0RgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei"
}
},
{
"type": "interim",
"title": "Simplify $$432180\\cdot\\:\\frac{x^{8}}{8}:{\\quad}\\frac{108045x^{8}}{2}$$",
"input": "432180\\cdot\\:\\frac{x^{8}}{8}",
"result": "=\\frac{108045x^{8}}{2}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{x^{8}\\cdot\\:432180}{8}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$4$$",
"result": "=\\frac{108045x^{8}}{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OawR7gxv5yRQ/29yKMZjhDQL3n3fhOyTG1gNZF/ic+8tOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOFFid9kS7S0Lngme8Z4CNG8RztKE552b3ssyeALQtWRjHimBRYRqHSWeJkuUPhfTCybZQzEaQAPEVeSyZLOxjdeRlRkNCFj6lzDp5km0ZOblwUbICdu7fNnZeePo9nSUFg=="
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "interim",
"title": "$$\\int\\:352947x^{6}dx=50421x^{7}$$",
"input": "\\int\\:352947x^{6}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=352947\\cdot\\:\\int\\:x^{6}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{6}dx",
"result": "=352947\\cdot\\:\\frac{x^{7}}{7}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{6+1}}{6+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{6+1}}{6+1}:{\\quad}\\frac{x^{7}}{7}$$",
"input": "\\frac{x^{6+1}}{6+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$6+1=7$$",
"result": "=\\frac{x^{7}}{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{7}}{7}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71PX4w684PGoRK55u2wyT0mo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ocYj7vVG0kubj3kttWOlD7CgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei"
}
},
{
"type": "interim",
"title": "Simplify $$352947\\cdot\\:\\frac{x^{7}}{7}:{\\quad}50421x^{7}$$",
"input": "352947\\cdot\\:\\frac{x^{7}}{7}",
"result": "=50421x^{7}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{x^{7}\\cdot\\:352947}{7}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{352947}{7}=50421$$",
"result": "=50421x^{7}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rwhTHRXh81mOPVPqr9fsZgz/KBCq4ROtt4G5r8ims7MtOtZYwUjyXhDTsNnn6ElrXRo3cTbMlTR//bt7PSYQaFBgSc24REh+dRFy9C9CZxUeNvb7k0sVmuwf19w9aD9Nul7pkAHpLoXPkSb8XK59w0Zr+fcWHMPd5IYu7MDfRKFlGjRtVn5wHRbJlFmiFdcy"
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "interim",
"title": "$$\\int\\:117649x^{5}dx=\\frac{117649x^{6}}{6}$$",
"input": "\\int\\:117649x^{5}dx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=117649\\cdot\\:\\int\\:x^{5}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{5}dx",
"result": "=117649\\cdot\\:\\frac{x^{6}}{6}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{5+1}}{5+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{5+1}}{5+1}:{\\quad}\\frac{x^{6}}{6}$$",
"input": "\\frac{x^{5+1}}{5+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$5+1=6$$",
"result": "=\\frac{x^{6}}{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{6}}{6}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70HUskMn2V8ZBlYunFrfJf6o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ocymAVtGK/9bBXqLD28AmPngQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei"
}
},
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{117649x^{6}}{6}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "step",
"result": "=\\frac{243x^{12}}{2}-1701x^{11}+\\frac{19845x^{10}}{2}-30870x^{9}+\\frac{108045x^{8}}{2}-50421x^{7}+\\frac{117649x^{6}}{6}"
},
{
"type": "step",
"primary": "Add a constant to the solution",
"result": "=\\frac{243x^{12}}{2}-1701x^{11}+\\frac{19845x^{10}}{2}-30870x^{9}+\\frac{108045x^{8}}{2}-50421x^{7}+\\frac{117649x^{6}}{6}+C",
"meta": {
"title": {
"extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$"
}
}
}
],
"meta": {
"solvingClass": "Integrals",
"practiceLink": "/practice/integration-practice#area=main&subtopic=Sum%20Rule",
"practiceTopic": "Integral Sum Rule"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=\\frac{243x^{12}}{2}-1701x^{11}+\\frac{19845x^{10}}{2}-30870x^{9}+\\frac{108045x^{8}}{2}-50421x^{7}+\\frac{117649x^{6}}{6}+C"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
Solution
Solution steps
Expand
Apply the Sum Rule:
Add a constant to the solution