Question

simplify {5(8^1/3+27^1/3)^1/3}^1/4

Answer 1

Answer:

5(5^1/3^1/4)

=5(50.759836)

=(5)(3.397053)

=16.985266

Answer 2

Answer:

$\left(5\left(8^{\dfrac{1}{3}}+27^{\dfrac{1}{3}}\right)^{\dfrac{1}{3}}\right)^{\dfrac{1}{4}}=5^{\dfrac{1}{3}}\quad \left(\mathrm{Decimal:\quad }\:1.70998\dots \right)$

Step-by-step explanation:

$\left(5\left(8^{\dfrac{1}{3}}+27^{\dfrac{1}{3}}\right)^{\dfrac{1}{3}}\right)^{\dfrac{1}{4}}$

$\left(8^{\dfrac{1}{3}}+27^{\dfrac{1}{3}}\right)^{\dfrac{1}{3}}$

$8^{\dfrac{1}{3}}$

$\mathrm{Factor\:the\:number:\:}\:8=2^3$

$=\left(2^3\right)^{\dfrac{1}{3}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(2^3\right)^{\dfrac{1}{3}}=2^{3\cdot \dfrac{1}{3}}=2$

$=2$

$27^{\dfrac{1}{3}}$

$\mathrm{Factor\:the\:number:\:}\:27=3^3$

$=\left(3^3\right)^{\dfrac{1}{3}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(3^3\right)^{\dfrac{1}{3}}=3^{3\cdot \dfrac{1}{3}}=3$

$=3$

$=\left(2+3\right)^{\dfrac{1}{3}}$

$\mathrm{Add\:the\:numbers:}\:2+3=5$

$=5^{\dfrac{1}{3}}$

$=\left(5\cdot \:5^{\dfrac{1}{3}}\right)^{\dfrac{1}{4}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a\cdot \:b\right)^n=a^nb^n$

$=5^{\dfrac{1}{4}}\left(5^{\dfrac{1}{3}}\right)^{\dfrac{1}{4}}$

$\left(5^{\dfrac{1}{3}}\right)^{\dfrac{1}{4}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$=5^{\dfrac{1}{3}\cdot \dfrac{1}{4}}$

$\dfrac{1}{3}\cdot \dfrac{1}{4}$

$\mathrm{Multiply\:fractions}:\quad \dfrac{a}{b}\cdot \dfrac{c}{d}=\dfrac{a\:\cdot \:c}{b\:\cdot \:d}$

$=\dfrac{1\cdot \:1}{3\cdot \:4}$

$\mathrm{Multiply\:the\:numbers:}\:1\cdot \:1=1$

$=\dfrac{1}{3\cdot \:4}$

$\mathrm{Multiply\:the\:numbers:}\:3\cdot \:4=12$

$=\dfrac{1}{12}$

$=5^{\dfrac{1}{12}}$

$=5^{\dfrac{1}{4}}\cdot \:5^{\dfrac{1}{12}}$

$\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}$

$5^{\dfrac{1}{4}}\cdot \:5^{\dfrac{1}{12}}=\:5^{\dfrac{1}{4}+\dfrac{1}{12}}$

$=5^{\dfrac{1}{4}+\dfrac{1}{12}}$

$\mathrm{Join}\:\dfrac{1}{4}+\dfrac{1}{12}$

$\dfrac{1}{4}+\dfrac{1}{12}$

$\mathrm{Least\:Common\:Multiplier\:of\:}4,\:12$

$4,\:12$

$\mathrm{Least\:Common\:Multiplier\:\left(LCM\right)}$

$\mathrm{The\:LCM\:of\:}a,\:b\mathrm{\:is\:the\:smallest\:positive\:number\:that\:is\:divisible\:by\:both\:}a\mathrm{\:and\:}b$

$\mathrm{Prime\:factorization\:of\:}4:\quad 2\cdot \:2$

$\mathrm{Prime\:factorization\:of\:}12:\quad 2\cdot \:2\cdot \:3$

$\mathrm{Multiply\:each\:factor\:the\:greatest\:number\:of\:times\:it\:occurs\:in\:either\:}4\mathrm{\:or\:}12$

$=2\cdot \:2\cdot \:3$

$\mathrm{Multiply\:the\:numbers:}\:2\cdot \:2\cdot \:3=12$

$=12$

$\mathrm{Adjust\:Fractions\:based\:on\:the\:LCM}$

$\mathrm{Multiply\:each\:numerator\:by\:the\:same\:amount\:needed\:to\:multiply\:its}$

$\mathrm{corresponding\:denominator\:to\:turn\:it\:into\:the\:LCM}\:12$

$\mathrm{For}\:\dfrac{1}{4}:\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:3$

$\dfrac{1}{4}=\dfrac{1\cdot \:3}{4\cdot \:3}=\dfrac{3}{12}$

$=\dfrac{3}{12}+\dfrac{1}{12}$

$\mathrm{Add\:the\:numbers:}\:3+1=4$

$=\dfrac{4}{12}$

$\mathrm{Cancel\:the\:common\:factor:}\:4$

$=\dfrac{1}{3}$

$=5^{\dfrac{1}{3}}$