Question

select the expression that is equivalent to \\(sqrt3{-125x^6y^5}\\).

answer
\\(5ix^2y^{\frac{5}{3}}\\) \\(-5x^3y^2\\)
\\(-5x^2y^{\frac{5}{3}}\\) \\(5ix^3y^2\\)

Explanation:

Step1: Analyze the cube root of -125

The cube root of -125 is -5 because \((-5)^3 = -125\).

Step2: Analyze the cube root of \(x^6\)

Using the property \(\sqrt[3]{a^n}=a^{\frac{n}{3}}\), for \(x^6\), we have \(\sqrt[3]{x^6}=x^{\frac{6}{3}} = x^2\).

Step3: Analyze the cube root of \(y^5\)

Using the same property, \(\sqrt[3]{y^5}=y^{\frac{5}{3}}\).

Step4: Combine the results

Putting it all together, \(\sqrt[3]{-125x^6y^5}=\sqrt[3]{-125}\times\sqrt[3]{x^6}\times\sqrt[3]{y^5}=-5x^2y^{\frac{5}{3}}\).

Answer:

\(-5x^2y^{\frac{5}{3}}\) (corresponding to the option \(-5x^2y^{\frac{5}{3}}\))