To simplify the expression [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex], we can follow these steps systematically:
1. Simplify the fraction inside the square root:
[tex]\[
\frac{2160 x^8}{60 x^2}
\][/tex]
First, compute the numerical part:
[tex]\[
\frac{2160}{60}
\][/tex]
Dividing 2160 by 60:
[tex]\[
\frac{2160}{60} = 36
\][/tex]
Now, handle the variable part:
[tex]\[
\frac{x^8}{x^2}
\][/tex]
Using the properties of exponents, [tex]\(\frac{x^8}{x^2} = x^{8-2} = x^6\)[/tex].
So, the fraction simplifies to:
[tex]\[
\frac{2160 x^8}{60 x^2} = 36 x^6
\][/tex]
2. Simplify the expression under the square root:
[tex]\[
\sqrt{36 x^6}
\][/tex]
3. Separate the square root of the product:
Using the property of square roots, [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{36 x^6} = \sqrt{36} \cdot \sqrt{x^6}
\][/tex]
4. Compute the individual square roots:
[tex]\[
\sqrt{36} = 6
\][/tex]
For the second term, [tex]\( \sqrt{x^6} \)[/tex]:
Recall that [tex]\( \sqrt{x^6} = x^{6/2} = x^3 \)[/tex].
5. Combine the results:
[tex]\[
\sqrt{36 x^6} = 6 \cdot x^3 = 6x^3
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex] is [tex]\(6x^3\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{6 x^3}
\][/tex]
