Question 120761


*[Tex \LARGE \sqrt[3]{1000x^{6}y^5}] Start with the given expression



*[Tex \LARGE \left(1000x^{6}y^5\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((1000)^1x^6y^5\right)^{\frac{1}{3}}] Rewrite 1000 as {{{1000^1}}}



*[Tex \LARGE (1000)^{1\left(\frac{1}{3}\right)}x^{6\left(\frac{1}{3}\right)}y^{5\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 (1000)^{\frac{1}{3}}x^{\frac{6}{3}}y^{\frac{5}{3}}] Now multiply the exponents

 

*[Tex \LARGE (1000)^{\frac{1}{3}}x^{2}y^{\frac{5}{3}}] Reduce

 

*[Tex \LARGE \sqrt[3]{1000}x^{2}\sqrt[3]{y^{5}}] Now convert back to radical notation




*[Tex \LARGE 10x^{2}\sqrt[3]{y^{5}}] Take the cube root of 1000 to get 10

 

So *[Tex \LARGE \sqrt[3]{1000x^{6}y^5}] simplifies to *[Tex \LARGE 10x^{2}\sqrt[3]{y^{5}}]




In other words, *[Tex \LARGE \sqrt[3]{1000x^{6}y^5}=10x^{2}\sqrt[3]{y^{5}}]