simplify (27/8)^{4/3}

{ "query": { "display": "simplify $$\\left(\\frac{27}{8}\\right)^{\\frac{4}{3}}$$", "symbolab_question": "SIMPLIFY#simplify (\\frac{27}{8})^{\\frac{4}{3}}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{81}{16}", "decimal": "5.0625", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(\\frac{27}{8}\\right)^{\\frac{4}{3}}=\\frac{81}{16}$$", "input": "\\left(\\frac{27}{8}\\right)^{\\frac{4}{3}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{27^{\\frac{4}{3}}}{8^{\\frac{4}{3}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$27^{\\frac{4}{3}}=81$$", "input": "27^{\\frac{4}{3}}", "steps": [ { "type": "step", "primary": "Factor the number: $$27=3^{3}$$", "result": "=\\left(3^{3}\\right)^{\\frac{4}{3}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad\\:a\\ge0$$", "result": "=3^{3\\cdot\\:\\frac{4}{3}}" }, { "type": "interim", "title": "$$3\\cdot\\:\\frac{4}{3}=4$$", "input": "3\\cdot\\:\\frac{4}{3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3}{1}$$", "result": "=\\frac{3}{1}\\cdot\\:\\frac{4}{3}" }, { "type": "step", "primary": "Cross-cancel: $$3$$", "result": "=\\frac{4}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA7Bbxk6/waZqXsSrNkwOGzKrju+5Z51e/ZZSD3gRHwjBMacxU057F7zjv8nRcs+opFjLxBb7GnxOcfSgBEBJAkNfAAN5X1HMORLqlII/vJUM" } }, { "type": "step", "result": "=3^{4}" }, { "type": "step", "primary": "$$3^{4}=81$$", "result": "=81" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FSSLComalBooeVjBHrCBKMo2PetG46f0uQUsvmJOZ+qjkVi15I8rBefLi4Iyt2wrrDmT3r7cjdnLNPU1X9yCmNNoRE3DyGkxNvzR81biiJS9xGLQ+gr9cTp9H7o+ZwLa" } }, { "type": "step", "result": "=\\frac{81}{8^{\\frac{4}{3}}}" }, { "type": "interim", "title": "$$8^{\\frac{4}{3}}=16$$", "input": "8^{\\frac{4}{3}}", "steps": [ { "type": "step", "primary": "Factor the number: $$8=2^{3}$$", "result": "=\\left(2^{3}\\right)^{\\frac{4}{3}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad\\:a\\ge0$$", "result": "=2^{3\\cdot\\:\\frac{4}{3}}" }, { "type": "interim", "title": "$$3\\cdot\\:\\frac{4}{3}=4$$", "input": "3\\cdot\\:\\frac{4}{3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3}{1}$$", "result": "=\\frac{3}{1}\\cdot\\:\\frac{4}{3}" }, { "type": "step", "primary": "Cross-cancel: $$3$$", "result": "=\\frac{4}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CzXS+MvaOUdKL6IkXGMFA7Bbxk6/waZqXsSrNkwOGzKrju+5Z51e/ZZSD3gRHwjBMacxU057F7zjv8nRcs+opFjLxBb7GnxOcfSgBEBJAkNfAAN5X1HMORLqlII/vJUM" } }, { "type": "step", "result": "=2^{4}" }, { "type": "step", "primary": "$$2^{4}=16$$", "result": "=16" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ojTXNRDlQXjCn1TaV607nUV2IovRuj7SUtFTM1ebUgRjZrOM+Q8JEbZyOig8BtyyuS45q9+rdwD/FCopkZicBfHeaVNwgreOGJrh1kEQ3erawWkW96ZDM24iNPvQNJMN" } }, { "type": "step", "result": "=\\frac{81}{16}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }