To simplify the expression [tex]\(\sqrt{\frac{2160x^8}{60x^2}}\)[/tex], we need to follow these steps:
1. Simplify the fraction inside the square root:
[tex]\[
\frac{2160x^8}{60x^2}
\][/tex]
2. Simplify the numerical coefficient:
The fraction [tex]\(\frac{2160}{60}\)[/tex] simplifies to:
[tex]\[
\frac{2160}{60} = 36
\][/tex]
3. Simplify the power of [tex]\(x\)[/tex]:
For the term [tex]\(x\)[/tex], we have:
[tex]\[
\frac{x^8}{x^2} = x^{8-2} = x^6
\][/tex]
So the expression becomes:
[tex]\[
\frac{2160x^8}{60x^2} = 36x^6
\][/tex]
4. Take the square root of the expression:
[tex]\[
\sqrt{36x^6}
\][/tex]
5. Simplify the square root:
The square root of a product is the product of the square roots. Thus:
[tex]\[
\sqrt{36x^6} = \sqrt{36} \cdot \sqrt{x^6}
\][/tex]
We know that:
[tex]\[
\sqrt{36} = 6 \quad \text{and} \quad \sqrt{x^6} = x^3
\][/tex]
(since [tex]\(\sqrt{x^6} = x^{6/2} = x^3\)[/tex]).
6. Combine the results:
[tex]\[
\sqrt{36x^6} = 6x^3
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{\frac{2160x^8}{60x^2}}\)[/tex] is [tex]\(6x^3\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{6x^3}
\][/tex]
