simplify 3/(4((1/4)+2)^{5/2)}

{ "query": { "display": "simplify $$\\frac{3}{4\\left(\\left(\\frac{1}{4}\\right)+2\\right)^{\\frac{5}{2}}}$$", "symbolab_question": "SIMPLIFY#simplify \\frac{3}{4((\\frac{1}{4})+2)^{\\frac{5}{2}}}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{8}{81}", "decimal": "0.09876…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{3}{4\\left(\\left(\\frac{1}{4}\\right)+2\\right)^{\\frac{5}{2}}}=\\frac{8}{81}$$", "input": "\\frac{3}{4\\left(\\left(\\frac{1}{4}\\right)+2\\right)^{\\frac{5}{2}}}", "steps": [ { "type": "step", "primary": "Apply rule: $$\\left(a\\right)=a$$", "secondary": [ "$$\\left(\\frac{1}{4}\\right)=\\frac{1}{4}$$" ], "result": "=\\frac{3}{4\\left(\\frac{1}{4}+2\\right)^{\\frac{5}{2}}}" }, { "type": "interim", "title": "Simplify $$4\\left(\\frac{1}{4}+2\\right)^{\\frac{5}{2}}:{\\quad}\\frac{243}{8}$$", "input": "4\\left(\\frac{1}{4}+2\\right)^{\\frac{5}{2}}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{4}+2\\right)^{\\frac{5}{2}}=\\frac{243}{32}$$", "input": "\\left(\\frac{1}{4}+2\\right)^{\\frac{5}{2}}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{4}+2=\\frac{9}{4}$$", "input": "\\frac{1}{4}+2", "steps": [ { "type": "interim", "title": "Convert $$2\\:$$to fraction $$:{\\quad}\\frac{8}{4}$$", "input": "2", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$a=\\frac{ab}{b}$$", "result": "=\\frac{2\\cdot\\:4}{4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{8}{4}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Join Convert Element to Fractions Title 1Eq" } }, { "type": "step", "result": "=\\frac{1}{4}+\\frac{8}{4}" }, { "type": "interim", "title": "$$\\frac{1}{4}+\\frac{8}{4}=\\frac{9}{4}$$", "input": "\\frac{1}{4}+\\frac{8}{4}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}$$", "result": "=\\frac{1+8}{4}" }, { "type": "step", "primary": "Add the numbers: $$1+8=9$$", "result": "=\\frac{9}{4}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72S64wvM73kdcljoY0cQeaOnZWJwSedBEL54aqIdCxSTdd47a0hQ8flDbGsI5To1dWKLdGjPOFc4xYqcnS7gljFO1T0snFOPqKXL+S6MxVmm+bV13fO5wjD5qSVzE8md8cIXcuy2kcBsEm7wiTrBnSMRpUWZvLMnPtt9MhdaOA3o=" } }, { "type": "step", "result": "=\\frac{9}{4}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jiecsLVzxlKEOBi9E8ElcACWKUbvV6WK3fDUgFtg3Q/ysS6ztOAKKFAEXmiY4WQmjx6D0MZ1j6OqGR+hYrY8gVNbbqpyK7JQEZdATEJR51hSbbrZkgHymD+qgVD/wwCAuxq/qgrA5qWTcQ1KY33A1Q==" } }, { "type": "step", "result": "=\\left(\\frac{9}{4}\\right)^{\\frac{5}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{9^{\\frac{5}{2}}}{4^{\\frac{5}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$9^{\\frac{5}{2}}=243$$", "input": "9^{\\frac{5}{2}}", "steps": [ { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\left(3^{2}\\right)^{\\frac{5}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad\\:a\\ge0$$", "result": "=3^{2\\cdot\\:\\frac{5}{2}}" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{5}{2}=5$$", "input": "2\\cdot\\:\\frac{5}{2}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2}{1}$$", "result": "=\\frac{2}{1}\\cdot\\:\\frac{5}{2}" }, { "type": "step", "primary": "Cross-cancel: $$2$$", "result": "=\\frac{5}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "result": "=5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvihCbp8QXDCU679cm5kOoT/Orju+5Z51e/ZZSD3gRHwjBrXKeeFsYwhYvayG3pqRTgjQhqRfFhpwnpyuz2TS2M3xocfS476fym4bdzecdfyQX" } }, { "type": "step", "result": "=3^{5}" }, { "type": "step", "primary": "$$3^{5}=243$$", "result": "=243" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MRQKogcNEqaErcHQa8z6ukV2IovRuj7SUtFTM1ebUgSzsHBJV0oRhKqf7h8oBCgaUFbueNPnH7GV/Ll/pp7u1pzNF8kbhlPPuKTqk65dwraLy5KGaW0n454+ZNJuzfyK" } }, { "type": "step", "result": "=\\frac{243}{4^{\\frac{5}{2}}}" }, { "type": "interim", "title": "$$4^{\\frac{5}{2}}=32$$", "input": "4^{\\frac{5}{2}}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\left(2^{2}\\right)^{\\frac{5}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc},\\:\\quad\\:a\\ge0$$", "result": "=2^{2\\cdot\\:\\frac{5}{2}}" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{5}{2}=5$$", "input": "2\\cdot\\:\\frac{5}{2}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2}{1}$$", "result": "=\\frac{2}{1}\\cdot\\:\\frac{5}{2}" }, { "type": "step", "primary": "Cross-cancel: $$2$$", "result": "=\\frac{5}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "result": "=5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvihCbp8QXDCU679cm5kOoT/Orju+5Z51e/ZZSD3gRHwjBrXKeeFsYwhYvayG3pqRTgjQhqRfFhpwnpyuz2TS2M3xocfS476fym4bdzecdfyQX" } }, { "type": "step", "result": "=2^{5}" }, { "type": "step", "primary": "$$2^{5}=32$$", "result": "=32" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7a/u1rQ6VRw4mNGegApMAw0V2IovRuj7SUtFTM1ebUgSB5cQUIPnysMUN/Jtl/Xok5g5ol4Ple/TlcQ5gQvzs1Z4QdBsVrCh1Vkc9hVUz7v0wiEwCRHS3csF2J0wDbMnJ" } }, { "type": "step", "result": "=\\frac{243}{32}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vQyZy3do97uZo8KmUPHpJSkwhtHNuRVUF6VPC9vE04h8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcauo82UNY6AjMDWiUd3uRlLHZWoi4I5sjEGJQNMZ8pSzEnfTNYyUK3FMD9ip3wtxSAFisaqE//8d+1JQ/R1Wv/e2mYNIew2FWNlI5Q7ct65o0+" } }, { "type": "step", "result": "=4\\cdot\\:\\frac{243}{32}" }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{243}{32}=\\frac{243}{8}$$", "input": "4\\cdot\\:\\frac{243}{32}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$4=\\frac{4}{1}$$", "result": "=\\frac{4}{1}\\cdot\\:\\frac{243}{32}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{b}\\cdot\\:\\frac{c}{d}=\\frac{a\\:\\cdot\\:\\:c}{b\\:\\cdot\\:\\:d}$$", "result": "=\\frac{4\\cdot\\:243}{1\\cdot\\:32}" }, { "type": "interim", "title": "$$\\frac{4\\cdot\\:243}{1\\cdot\\:32}=\\frac{243}{8}$$", "input": "\\frac{4\\cdot\\:243}{1\\cdot\\:32}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:32=32$$", "result": "=\\frac{4\\cdot\\:243}{32}" }, { "type": "step", "primary": "Factor the number: $$32=4\\cdot\\:8$$", "result": "=\\frac{4\\cdot\\:243}{4\\cdot\\:8}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\frac{243}{8}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VwvNcHE64QyhrTV7sZdoif/eUrnxVaEv5oQIxCuCHx18kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcaunPoSvo7N6RmYCVMvwg0ybAHk1eLttM1tdVeTyzsC+lrYHaDtdC3DKjjdTXa5wPUqGmMcgcXHSjULURmBUQM6cBhTAkdIbckpmuWnf5dk4if" } }, { "type": "step", "result": "=\\frac{243}{8}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Solver2", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV89rCmUdgr0Uj5MHzC4qFG8oDnzlbPZjyKgy1eUCFsLd5THSJDwRnQ5uVyQcGkRXod76DMDH258ED6TCmtXj8VmqwpoZTWr3XWpe0sOSHOXj790iw2Mcxdrjccn68ZUxGNGFMCR0htySma5ad/l2TiJ8=" } }, { "type": "step", "result": "=\\frac{243}{8}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bRuMNnk0kgWe4IBgNnOroONUuiWAUwchVFM6CSi1c+lFdiKL0bo+0lLRUzNXm1IE/aL2Coo0GISQwm8bv5wDicjtMSaGE94EJ/8wFDZDuGtFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qnS3MJzpbcYq/VMrXBipFvwuu4/YnSDDbkSFK2DZmbcHsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=\\frac{3}{\\frac{243}{8}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{3\\cdot\\:8}{243}" }, { "type": "interim", "title": "Cancel $$\\frac{3\\cdot\\:8}{243}:{\\quad}\\frac{8}{81}$$", "input": "\\frac{3\\cdot\\:8}{243}", "steps": [ { "type": "step", "primary": "Factor the number: $$243=3\\cdot\\:81$$", "result": "=\\frac{3\\cdot\\:8}{3\\cdot\\:81}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{8}{81}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYseXde4en/Yhb7i4KBRjo6AAlilG71elit3w1IBbYN0PeH+rcvZk9LuAUQZeZZ0NvIpDLDOgwPf6olWv7k47cPN6pfF1z6umzUJTJvt+ojYZ3Rhao/hEKgGYcV+JSwxeA/qUy+YTdHwjZFH2zNqudlW/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "=\\frac{8}{81}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }