Question

If x>0 and y>0, which expression is equivalent to ^3 √ 1024x^15y^14

Answer 1

The given expression is;

`\sqrt[3]{1024x^(15)y^(14)}`

We need to simplify the expression for x >0 and y > 0

Split each term of the given expression as;

`\sqrt[3]{1024x^(15)y^(14)}=\sqrt[3]{1024}\sqrt[3]{x^(15)}\sqrt[3]{y^(14)}`

Simplify each term;

`\begin{gathered} \sqrt[3]{1024}=\sqrt[3]{2*8*8*8} \\ \sqrt[3]{1024}=8\sqrt[3]{2} \end{gathered}`

Now for,

`\begin{gathered} \sqrt[3]{x^(15)} \\ \sqrt[3]{x^(15)}=(x)^{(15)/(3)} \\ \sqrt[3]{x^(15)}=x^5 \end{gathered}`

Now for the third expression,

`\begin{gathered} \sqrt[3]{y^(14)}=\sqrt[3]{y^(12)y^2}^{} \\ \sqrt[3]{y^(14)}=\sqrt[3]{y^(3*4)y^2} \\ \sqrt[3]{y^(14)}=\sqrt[3]{y^(12)}\sqrt[3]{y^2} \\ \sqrt[3]{y^(14)}=(y)^{(12)/(3)}^{}\sqrt[3]{y^2} \\ \sqrt[3]{y^(14)}=y^4^{}\sqrt[3]{y^2} \end{gathered}`

Substitute these value and solve for x;

`\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}`

Answer : C)

`8x^5y^4\sqrt[3]{2y^2}`