Respuesta :
The given expression is;
[tex]\sqrt[3]{1024x^{15}y^{14}}[/tex]We need to simplify the expression for x >0 and y > 0
Split each term of the given expression as;
[tex]\sqrt[3]{1024x^{15}y^{14}}=\sqrt[3]{1024}\sqrt[3]{x^{15}}\sqrt[3]{y^{14}}[/tex]Simplify each term;
[tex]\begin{gathered} \sqrt[3]{1024}=\sqrt[3]{2\times8\times8\times8} \\ \sqrt[3]{1024}=8\sqrt[3]{2} \end{gathered}[/tex]Now for,
[tex]\begin{gathered} \sqrt[3]{x^{15}} \\ \sqrt[3]{x^{15}}=(x)^{\frac{15}{3}} \\ \sqrt[3]{x^{15}}=x^5 \end{gathered}[/tex]Now for the third expression,
[tex]\begin{gathered} \sqrt[3]{y^{14}}=\sqrt[3]{y^{12}y^2}^{} \\ \sqrt[3]{y^{14}}=\sqrt[3]{y^{3\times4}y^2} \\ \sqrt[3]{y^{14}}=\sqrt[3]{y^{12}}\sqrt[3]{y^2} \\ \sqrt[3]{y^{14}}=(y)^{\frac{12}{3}}^{}\sqrt[3]{y^2} \\ \sqrt[3]{y^{14}}=y^4^{}\sqrt[3]{y^2} \end{gathered}[/tex]Substitute these value and solve for x;
[tex]\begin{gathered} \sqrt[3]{1024x^{15}y^{14}}=\sqrt[3]{1024}\sqrt[3]{x^{15}}\sqrt[3]{y^{14}} \\ \sqrt[3]{1024x^{15}y^{14}}=8\sqrt[3]{2}x^5y^4\sqrt[3]{y^2} \\ \sqrt[3]{1024x^{15}y^{14}}=8x^5y^4\sqrt[3]{2y^2} \end{gathered}[/tex]Answer : C)
[tex]8x^5y^4\sqrt[3]{2y^2}[/tex]
