2tan(15)

{ "query": { "display": "$$2\\tan\\left(15^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#2\\tan(15^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "2(2-\\sqrt{3})", "decimal": "0.53589…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\tan\\left(15^{\\circ\\:}\\right)=2\\left(2-\\sqrt{3}\\right)$$", "input": "2\\tan\\left(15^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\tan\\left(15^{\\circ\\:}\\right)=2-\\sqrt{3}$$", "input": "\\tan\\left(15^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\frac{\\tan\\left(45^{\\circ\\:}\\right)-\\tan\\left(30^{\\circ\\:}\\right)}{1+\\tan\\left(45^{\\circ\\:}\\right)\\tan\\left(30^{\\circ\\:}\\right)}$$", "input": "\\tan\\left(15^{\\circ\\:}\\right)", "result": "=\\frac{\\tan\\left(45^{\\circ\\:}\\right)-\\tan\\left(30^{\\circ\\:}\\right)}{1+\\tan\\left(45^{\\circ\\:}\\right)\\tan\\left(30^{\\circ\\:}\\right)}", "steps": [ { "type": "step", "primary": "Write $$\\tan\\left(15^{\\circ\\:}\\right)\\:$$as $$\\tan\\left(45^{\\circ\\:}-30^{\\circ\\:}\\right)$$", "result": "=\\tan\\left(45^{\\circ\\:}-30^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Use the Angle Difference identity: $$\\tan\\left(s-t\\right)=\\frac{\\tan\\left(s\\right)-\\tan\\left(t\\right)}{1+\\tan\\left(s\\right)\\tan\\left(t\\right)}$$", "result": "=\\frac{\\tan\\left(45^{\\circ\\:}\\right)-\\tan\\left(30^{\\circ\\:}\\right)}{1+\\tan\\left(45^{\\circ\\:}\\right)\\tan\\left(30^{\\circ\\:}\\right)}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\tan\\left(45^{\\circ\\:}\\right)=1$$", "input": "\\tan\\left(45^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\tan\\left(30^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{3}$$", "input": "\\tan\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{1-\\frac{\\sqrt{3}}{3}}{1+1\\cdot\\:\\frac{\\sqrt{3}}{3}}" }, { "type": "interim", "title": "Simplify $$\\frac{1-\\frac{\\sqrt{3}}{3}}{1+1\\cdot\\:\\frac{\\sqrt{3}}{3}}:{\\quad}2-\\sqrt{3}$$", "input": "\\frac{1-\\frac{\\sqrt{3}}{3}}{1+1\\cdot\\:\\frac{\\sqrt{3}}{3}}", "result": "=2-\\sqrt{3}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{\\sqrt{3}}{3}=\\frac{\\sqrt{3}}{3}$$", "result": "=\\frac{1-\\frac{\\sqrt{3}}{3}}{1+\\frac{\\sqrt{3}}{3}}" }, { "type": "interim", "title": "Join $$1+\\frac{\\sqrt{3}}{3}:{\\quad}\\frac{\\sqrt{3}+1}{\\sqrt{3}}$$", "input": "1+\\frac{\\sqrt{3}}{3}", "result": "=\\frac{1-\\frac{\\sqrt{3}}{3}}{\\frac{\\sqrt{3}+1}{\\sqrt{3}}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=\\frac{1\\cdot\\:3}{3}+\\frac{\\sqrt{3}}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3+\\sqrt{3}}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3+\\sqrt{3}}{3}" }, { "type": "interim", "title": "Factor $$3+\\sqrt{3}:{\\quad}\\sqrt{3}\\left(\\sqrt{3}+1\\right)$$", "input": "3+\\sqrt{3}", "result": "=\\frac{\\sqrt{3}\\left(\\sqrt{3}+1\\right)}{3}", "steps": [ { "type": "step", "primary": "$$3=\\sqrt{3}\\sqrt{3}$$", "result": "=\\sqrt{3}\\sqrt{3}+\\sqrt{3}" }, { "type": "step", "primary": "Factor out common term $$\\sqrt{3}$$", "result": "=\\sqrt{3}\\left(\\sqrt{3}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{3}\\left(\\sqrt{3}+1\\right)}{3}:{\\quad}\\frac{\\sqrt{3}+1}{\\sqrt{3}}$$", "input": "\\frac{\\sqrt{3}\\left(\\sqrt{3}+1\\right)}{3}", "result": "=\\frac{\\sqrt{3}+1}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3^{\\frac{1}{2}}\\left(1+\\sqrt{3}\\right)}{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{3^{\\frac{1}{2}}}{3^{1}}=\\frac{1}{3^{1-\\frac{1}{2}}}$$" ], "result": "=\\frac{\\sqrt{3}+1}{3^{1-\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{\\sqrt{3}+1}{3^{\\frac{1}{2}}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$3^{\\frac{1}{2}}=\\sqrt{3}$$" ], "result": "=\\frac{\\sqrt{3}+1}{\\sqrt{3}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnfh1Ad0IEPQvc2ESPB5puRfGSbnHF3dy2IBE6JLg61g3XeO2tIUPH5Q2xrCOU6NXWH9vsh/4pLjp0PVztdqCI/r6fypiX4xioilkqMghC0RN3W8Wbfems81x5yFZ9Q32ehxHvj7Dx5XU5+l2MpjOSEcWiVTs17fc7EVtPvOUl6gb4VVQMAjS9zWi3yHD36nYcULWHg5m51XzIQExbFCFYo=" } } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Join $$1-\\frac{\\sqrt{3}}{3}:{\\quad}\\frac{\\sqrt{3}-1}{\\sqrt{3}}$$", "input": "1-\\frac{\\sqrt{3}}{3}", "result": "=\\frac{\\frac{\\sqrt{3}-1}{\\sqrt{3}}}{\\frac{\\sqrt{3}+1}{\\sqrt{3}}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=\\frac{1\\cdot\\:3}{3}-\\frac{\\sqrt{3}}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3-\\sqrt{3}}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3-\\sqrt{3}}{3}" }, { "type": "interim", "title": "Factor $$3-\\sqrt{3}:{\\quad}\\sqrt{3}\\left(\\sqrt{3}-1\\right)$$", "input": "3-\\sqrt{3}", "result": "=\\frac{\\sqrt{3}\\left(\\sqrt{3}-1\\right)}{3}", "steps": [ { "type": "step", "primary": "$$3=\\sqrt{3}\\sqrt{3}$$", "result": "=\\sqrt{3}\\sqrt{3}-\\sqrt{3}" }, { "type": "step", "primary": "Factor out common term $$\\sqrt{3}$$", "result": "=\\sqrt{3}\\left(\\sqrt{3}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{3}\\left(\\sqrt{3}-1\\right)}{3}:{\\quad}\\frac{\\sqrt{3}-1}{\\sqrt{3}}$$", "input": "\\frac{\\sqrt{3}\\left(\\sqrt{3}-1\\right)}{3}", "result": "=\\frac{\\sqrt{3}-1}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3^{\\frac{1}{2}}\\left(\\sqrt{3}-1\\right)}{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{3^{\\frac{1}{2}}}{3^{1}}=\\frac{1}{3^{1-\\frac{1}{2}}}$$" ], "result": "=\\frac{\\sqrt{3}-1}{3^{1-\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{\\sqrt{3}-1}{3^{\\frac{1}{2}}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$3^{\\frac{1}{2}}=\\sqrt{3}$$" ], "result": "=\\frac{\\sqrt{3}-1}{\\sqrt{3}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnfh1Ad0IEPQvc2ESPB5puTtY5m62xb0CG373ScqTe6k3XeO2tIUPH5Q2xrCOU6NXWH9vsh/4pLjp0PVztdqCI/DJLRYcpZBet9iUiAnRECQN3W8Wbfems81x5yFZ9Q32ehxHvj7Dx5XU5+l2MpjOSEcWiVTs17fc7EVtPvOUl6gJOgYPE94ZakRHl084sk7A8ULWHg5m51XzIQExbFCFYo=" } } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$", "result": "=\\frac{\\left(\\sqrt{3}-1\\right)\\sqrt{3}}{\\sqrt{3}\\left(\\sqrt{3}+1\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=\\frac{\\sqrt{3}-1}{\\sqrt{3}+1}" }, { "type": "interim", "title": "Rationalize $$\\frac{\\sqrt{3}-1}{\\sqrt{3}+1}:{\\quad}2-\\sqrt{3}$$", "input": "\\frac{\\sqrt{3}-1}{\\sqrt{3}+1}", "result": "=2-\\sqrt{3}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}-1}{\\sqrt{3}-1}$$", "result": "=\\frac{\\left(\\sqrt{3}-1\\right)\\left(\\sqrt{3}-1\\right)}{\\left(\\sqrt{3}+1\\right)\\left(\\sqrt{3}-1\\right)}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}+1$$" } } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}-1\\right)\\left(\\sqrt{3}-1\\right)=4-2\\sqrt{3}$$", "input": "\\left(\\sqrt{3}-1\\right)\\left(\\sqrt{3}-1\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\left(\\sqrt{3}-1\\right)\\left(\\sqrt{3}-1\\right)=\\:\\left(\\sqrt{3}-1\\right)^{1+1}$$" ], "result": "=\\left(\\sqrt{3}-1\\right)^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\left(\\sqrt{3}-1\\right)^{2}" }, { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=\\sqrt{3},\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=\\left(\\sqrt{3}\\right)^{2}-2\\sqrt{3}\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$\\left(\\sqrt{3}\\right)^{2}-2\\sqrt{3}\\cdot\\:1+1^{2}:{\\quad}4-2\\sqrt{3}$$", "input": "\\left(\\sqrt{3}\\right)^{2}-2\\sqrt{3}\\cdot\\:1+1^{2}", "result": "=4-2\\sqrt{3}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\left(\\sqrt{3}\\right)^{2}-2\\cdot\\:1\\cdot\\:\\sqrt{3}+1" }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}=3$$", "input": "\\left(\\sqrt{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XEzlndel0x/QlMW8VTD5UXyRHuGw7+tM5METTDj6vVEMSTe+hEEEr2+K9b3W9JkG8SrqrDW4mFcEK+hPNqZN8jCFrm8vCvLZxZsdY4NLQVuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$2\\sqrt{3}\\cdot\\:1=2\\sqrt{3}$$", "input": "2\\sqrt{3}\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78DwjsdMBjp9zBvqajFEcPGBFH3ZqAOJQyqKKX506iE3MwViaLUXkeD+JukROhWdj8RyasLmvIudhnO8XBHEAXy3qcDn8lI3dwHZacS+mAY/NxfbGzb+lvIPq3MFWNFHA/uXNPrOdbPfrHfRFrUo5Ug==" } }, { "type": "step", "result": "=3-2\\sqrt{3}+1" }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4-2\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72Tj1eRE79IZYtdTGOh0xx9kNCRgnJ9W/VqpP5HaKI14gJ/ZZA32ZInFBpDtxBfiKW/PSj9bSe2W83KFyKPnYa7m0T+jD13GzAl2W1XYHsNfMpfN7WBEZJCvLPqWdHo6cGJOaDx6GdXRxfS3R6ikn3P+b8Q9uTDfpmc2YQ5kZTDE=" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}+1\\right)\\left(\\sqrt{3}-1\\right)=2$$", "input": "\\left(\\sqrt{3}+1\\right)\\left(\\sqrt{3}-1\\right)", "result": "=\\frac{4-2\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$\\left(a+b\\right)\\left(a-b\\right)=a^{2}-b^{2}$$", "secondary": [ "$$a=\\sqrt{3},\\:b=1$$" ], "result": "=\\left(\\sqrt{3}\\right)^{2}-1^{2}", "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Expand Difference of Squares" } }, { "type": "interim", "title": "Simplify $$\\left(\\sqrt{3}\\right)^{2}-1^{2}:{\\quad}2$$", "input": "\\left(\\sqrt{3}\\right)^{2}-1^{2}", "result": "=2", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\left(\\sqrt{3}\\right)^{2}-1" }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}=3$$", "input": "\\left(\\sqrt{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XEzlndel0x/QlMW8VTD5UXyRHuGw7+tM5METTDj6vVEMSTe+hEEEr2+K9b3W9JkG8SrqrDW4mFcEK+hPNqZN8jCFrm8vCvLZxZsdY4NLQVuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=3-1" }, { "type": "step", "primary": "Subtract the numbers: $$3-1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7U0zW5yAqJTQxerKmrcJ6INkNCRgnJ9W/VqpP5HaKI14gJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUB9FohUoLgFEfrgLg8MRrO3o2ypjHn+SP5inl8xZKAavAfJr86b6/Uwq/vO4okaMkQ==" } }, { "type": "interim", "title": "Factor $$4-2\\sqrt{3}:{\\quad}2\\left(2-\\sqrt{3}\\right)$$", "input": "4-2\\sqrt{3}", "result": "=\\frac{2\\left(2-\\sqrt{3}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:2-2\\sqrt{3}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(2-\\sqrt{3}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=2-\\sqrt{3}" } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PndYYYIot5Qf6JPEzqJ6WN96/HZdAquz40rqyWWEyqNjrM3bkBb38YTIZq+4TTCM33r8dl0Cq7PjSurJZYTKowCWKUbvV6WK3fDUgFtg3Q9Q8N5AhvF3fqDx1npIGFrNhUulnmFIQQyhpb8h/WI6dHql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86H0uSW7nx02QXmeDWZDikF/eIruuCIjus9+fS4UUJcXO/mA7jT8tN5uQ7OP9M7qv/1PzpCkyT1QTa1+n32P3NGPg==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=2\\left(2-\\sqrt{3}\\right)" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }