Question 211272
First, let's simplify {{{sqrt(48x^8)}}}



{{{sqrt(48*x^8)}}} Start with the given expression.



{{{sqrt(16*3*x^8)}}} Factor {{{48}}} into {{{16*3}}}



{{{sqrt(16*3*x^2*x^2*x^2*x^2)}}} Factor {{{x^8}}} into {{{x^2*x^2*x^2*x^2}}}



{{{sqrt(16)*sqrt(3)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)}}} Break up the square root using the identity {{{sqrt(A*B)=sqrt(A)*sqrt(B)}}}.



{{{4*sqrt(3)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)}}} Take the square root of {{{16}}} to get {{{4}}}.



{{{4*sqrt(3)*x*x*x*x}}} Take the square root of {{{x^2}}} to get {{{x}}}.



{{{4x^4*sqrt(3)}}} Rearrange and multiply the terms.



So {{{sqrt(48*x^8)}}} simplifies to {{{4x^4*sqrt(3)}}}



In other words, {{{sqrt(48*x^8)=4x^4*sqrt(3)}}} where every variable is non-negative.



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So {{{root(3,sqrt(48*x^8))}}} simplifies to {{{root(3,4x^4*sqrt(3))}}}



Ie. {{{root(3,sqrt(48*x^8))=root(3,4x^4*sqrt(3))}}} where {{{x>=0}}}



Now let's simplify {{{root(3,4x^4*sqrt(3))}}}



{{{root(3,4x^4*sqrt(3))}}} Start with the given expression.



{{{root(3,4x^4)*root(3,sqrt(3))}}} Break up the root.



{{{root(3,4x*x^3)*root(3,sqrt(3))}}} Factor {{{x^4}}} to get {{{x*x^3}}}



{{{root(3,4x)*root(3,x^3)*root(3,sqrt(3))}}} Break up the first root.



{{{root(3,4x)*x*root(3,sqrt(3))}}} Take the cube root of {{{x^3}}} to get {{{x}}}



{{{(4x)^(1/3)^""*x*(3^(1/2))^(1/3)}}} Convert to exponential notation.



{{{(4x)^(1/3)*x*3^(1/6)^""}}} Multiply the exponents.



{{{(4x)^(2/6)^""*x*3^(1/6)}}} Rewrite the first exponent {{{1/3}}} as {{{2/6}}}



{{{x*((4x)^(2/6)^""*3^(1/6))}}} Rearrange the terms.



{{{x*((4x)^2*3^1)^(1/6)}}} Factor out {{{1/6}}}



{{{x*(16x^2*3)^(1/6)}}} Square and simplify



{{{x*(48x^2)^(1/6)}}} Multiply



{{{x*root(6,48x^2)}}} Convert back to radical notation.




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Answer:



So {{{root(3,sqrt(48*x^8))}}} completely simplifies to {{{x*root(6,48x^2)}}}



In other words, {{{root(3,sqrt(48*x^8))=x*root(6,48x^2)}}} where {{{x>=0}}}