simplify ((2a^4)/(7b^5))^6

{ "query": { "display": "simplify $$\\left(\\frac{2a^{4}}{7b^{5}}\\right)^{6}$$", "symbolab_question": "SIMPLIFY#simplify (\\frac{2a^{4}}{7b^{5}})^{6}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{64a^{24}}{117649b^{30}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Simplify $$\\left(\\frac{2a^{4}}{7b^{5}}\\right)^{6}:{\\quad}\\frac{64a^{24}}{117649b^{30}}$$", "input": "\\left(\\frac{2a^{4}}{7b^{5}}\\right)^{6}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(2a^{4}\\right)^{6}}{\\left(7b^{5}\\right)^{6}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$\\left(2a^{4}\\right)^{6}:{\\quad}64a^{24}$$", "input": "\\left(2a^{4}\\right)^{6}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(2a^{4}\\right)^{6}=2^{6}\\left(a^{4}\\right)^{6}$$" ], "result": "=2^{6}\\left(a^{4}\\right)^{6}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{6}=64$$", "result": "=64\\left(a^{4}\\right)^{6}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad$$ assuming $$a\\ge0$$", "secondary": [ "$$\\left(a^{4}\\right)^{6}=a^{4\\cdot\\:6}$$" ], "result": "=64a^{4\\cdot\\:6}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:6=24$$", "result": "=64a^{24}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uvV9pKFfo+C3Ez2OuJSiGVXTSum/z5kLpMzXS1UJIezBu7sDNjbOTxvhNf+s8wUWzFkijuHUjV2f2yEpoNqosu5AIz++qluupTlLFEcE9J0EephL+1qMtWux9VaZ1YyAdS/SPBT3T9nPUaFhJbe4gQ==" } }, { "type": "step", "result": "=\\frac{64a^{24}}{\\left(7b^{5}\\right)^{6}}" }, { "type": "interim", "title": "Simplify $$\\left(7b^{5}\\right)^{6}:{\\quad}117649b^{30}$$", "input": "\\left(7b^{5}\\right)^{6}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(7b^{5}\\right)^{6}=7^{6}\\left(b^{5}\\right)^{6}$$" ], "result": "=7^{6}\\left(b^{5}\\right)^{6}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Simplify $$\\left(b^{5}\\right)^{6}:{\\quad}b^{30}$$", "input": "\\left(b^{5}\\right)^{6}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad$$ assuming $$a\\ge0$$", "result": "=b^{5\\cdot\\:6}" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:6=30$$", "result": "=b^{30}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7we3L/tB987hT/b9B3VLJ396GQqufR6tr2vPxOUv7H+9ji4mJUr7EjjoMQPbcKlX6P8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXSljrvT6l9o29qZuyJLSr2fJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=7^{6}b^{30}" }, { "type": "step", "primary": "$$7^{6}=117649$$", "result": "=117649b^{30}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7m59F3fcd5pKD3fAI3WN+bFXTSum/z5kLpMzXS1UJIexyg8xapfCvLFBGgk2RcNIult56zebsZigb0YXjtc2z5U3kCh3oevUunZ7/b0qFKBR4ZmRKAjKiQlOi50PwXZI9iqPI+/qlCiOVNLNgSgyMOQ==" } }, { "type": "step", "result": "=\\frac{64a^{24}}{117649b^{30}}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }