Question 126506
{{{root(3,162a^10b^5)/root(3,6a^8b^4)}}} Start with the given expression



First let's simplify the numerator {{{root(3,162a^10b^5)}}}



*[Tex \LARGE \sqrt[3]{162a^{10}b^5}] Start with the given expression



*[Tex \LARGE \sqrt[3]{162}\sqrt[3]{a^{10}}\sqrt[3]{b^{5}}] Break up the roots


{{{3*root(3,6)*a^3*root(3,a)*b*root(3,b^2)}}}   Simplify {{{root(3,162)}}} to get {{{3*root(3,6)}}}.  Simplify {{{root(3,a^10)}}} to get {{{a^3*root(3,a)}}}. Simplify {{{root(3,b^5)}}} to get {{{b*root(3,b^2)}}}




{{{3a^3b*root(3,6ab^2)}}}  Multiply and combine the roots



So  {{{root(3,162a^10b^5)}}} simplifies to {{{3a^3b*root(3,6ab^2)}}}


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Now let's simplify the denominator {{{root(3,6a^8b^4)}}}



*[Tex \LARGE \sqrt[3]{6a^8b^4}] Start with the given expression



*[Tex \LARGE \sqrt[3]{6}\sqrt[3]{a^8}\sqrt[3]{b^{4}}] Break up the roots


{{{root(3,6)*a^2*root(3,a^2)*b*root(3,b)}}}  Simplify {{{root(3,a^8)}}} to get {{{a^2*root(3,a^2)}}}. Simplify {{{root(3,b^4)}}} to get {{{b*root(3,b)}}}




{{{a^2b*root(3,6a^2b)}}}  Multiply and combine the roots


So {{{root(3,6a^8b^4)}}} simplifies to {{{a^2b*root(3,6a^2b)}}} 






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So now our expression becomes


{{{(3a^3b*root(3,6ab^2))/(a^2b*root(3,6a^2b))}}}




{{{(3a*root(3,6ab^2))/(root(3,6a^2b))}}} Divide {{{(3a^3b)/(a^2b)}}} to get {{{3a}}}



{{{3a*root(3,(6ab^2)/(6a^2b))}}} Combine the roots



{{{3a*root(3,b/a)}}} Divide {{{(6ab^2)/(6a^2b)}}} to get {{{b/a}}}





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Answer:



So {{{root(3,162a^10b^5)/root(3,6a^8b^4)}}}  simplifies to {{{3a*root(3,b/a)}}}




In other words, {{{root(3,162a^10b^5)/root(3,6a^8b^4)=3a*root(3,b/a)}}}