Question

select the expression that is equivalent to \\(\sqrt3{216x^{12}y^{15}}\\).

answer
\\(\bigcirc\\) \\(6x^4y^5\\) \\(\bigcirc\\) \\(72x^4y^5\\) \\(\bigcirc\\) \\(72x^9y^{12}\\) \\(\bigcirc\\) \\(6x^9y^{12}\\)

Explanation:

Step1: Simplify the cube root of 216

We know that \(6^3 = 216\), so \(\sqrt[3]{216}=6\).

Step2: Simplify the cube root of \(x^{12}\)

Using the property \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), for \(\sqrt[3]{x^{12}}\), we have \(x^{\frac{12}{3}} = x^4\).

Step3: Simplify the cube root of \(y^{15}\)

Using the same property \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), for \(\sqrt[3]{y^{15}}\), we have \(y^{\frac{15}{3}} = y^5\).

Step4: Combine the results

Multiplying the results from Step1, Step2, and Step3 together, we get \(\sqrt[3]{216x^{12}y^{15}} = 6x^4y^5\).

Answer:

\(6x^4y^5\) (corresponding to the option: \(6x^4y^5\))