Question 194024
{{{root(3,xy^5)*root(3,x^(19)y^(17))}}} Start with the given expression



{{{root(3,(xy^5)(x^(19)y^(17)))}}} Combine the roots



{{{root(3,x^(1+19)y^(5+17))}}} Multiply by adding the exponents.



{{{root(3,x^(20)y^(22))}}} Add



*[Tex \LARGE \left(x^{20}y^{22}\right)^{\frac{1}{3}}] Convert to exponential notation



*[Tex \LARGE x^{20\left(\frac{1}{3}\right)}y^{22\left(\frac{1}{3}\right)}] Multiply the exponents.



*[Tex \LARGE x^{\frac{20}{3}}y^{\frac{22}{3}}] Multiply



*[Tex \LARGE x^{6\frac{2}{3}}y^{7\frac{1}{3}}] Convert the improper fractions to mixed fractions.


*[Tex \LARGE x^{6+\frac{2}{3}}y^{7+\frac{1}{3}}] Expand



*[Tex \LARGE x^6x^{\frac{2}{3}}y^7y^{\frac{1}{3}}] Break up the exponents using the identity {{{x^(y+z)=x^y*x^z}}}



*[Tex \LARGE x^6y^7\left(x^2y\right)^{\frac{1}{3}}] Rearrange the terms.



{{{x^6y^7*root(3,x^2y)}}} Convert back to radical notation



So {{{root(3,xy^5)*root(3,x^(19)y^(17))=x^6y^7*root(3,x^2y)}}}