Question 126389
{{{root(3,y^13)*root(3,81y^14)}}} Start with the given expression



{{{root(3,y^13*81y^14)}}} Combine the roots



{{{root(3,81y^27)}}} Multiply. Remember when you multiply like bases with exponents, you add the exponents.




*[Tex \LARGE \left(81y^{27}\right)^{\frac{1}{3}}] Convert the expression from radical notation to exponent notation. Remember *[Tex \LARGE \sqrt[3]{\textrm{A}}=\textrm{A}^{\frac{1}{3}}]



*[Tex \LARGE \left((81)^1y^27\right)^{\frac{1}{3}}] Rewrite 81 as {{{81^1}}}



*[Tex \LARGE (81)^{1\left(\frac{1}{3}\right)}y^{27\left(\frac{1}{3}\right)}] Now distribute the exponent Now distribute the outer exponent {{{1/3}}} to each exponent in the parenthesis. Remember {{{(x^y)^z=x^(y*z)}}}

 

*[Tex \LARGE (81)^{\frac{1}{3}}y^{\frac{27}{3}}] Now multiply the exponents

 

*[Tex \LARGE (81)^{\frac{1}{3}}y^{9}] Reduce

 

*[Tex \LARGE \sqrt[3]{81}y^{9}] Now convert back to radical notation


{{{root(3,27*3)y^9}}} Factor 81 into 27*3



{{{root(3,27)*root(3,3)y^9}}} Break up the root



{{{3*root(3,3)y^9}}} Take the third root of 27 to get 3 



{{{3y^9*root(3,3)}}} Multiply 


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Answer:


So {{{root(3,y^13)*root(3,81y^14)}}}  simplifies to {{{3y^9*root(3,3)}}}




In other words, {{{root(3,y^13)*root(3,81y^14)=3y^9*root(3,3)}}}