Question

which expression is equivalent to \\(\sqrt3{216x^6}\\)
\\(\circ\\) \\(6x^2\\)
\\(\circ\\) \\(6\sqrt3{x^6}\\)
\\(\circ\\) \\(5x^6\\)
\\(\circ\\) \\(25x^2\\)
\\(\circ\\) \\(5x^3\\)

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 \( x^6 \)

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

Step3: Combine the results

Using the property \( \sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b} \) (where \( a = 216 \) and \( b=x^6 \)), we get \( \sqrt[3]{216x^6}=\sqrt[3]{216}\cdot\sqrt[3]{x^6}=6\cdot x^2 = 6x^2 \).

Answer:

\( 6x^2 \) (the first option)