To simplify the expression [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex], we'll go through the process step by step:
1. Simplify the fraction inside the square root:
[tex]\[
\frac{2160 x^8}{60 x^2}
\][/tex]
2. Simplifying the numerical part:
[tex]\[
\frac{2160}{60}
\][/tex]
Perform the division:
[tex]\[
\frac{2160}{60} = 36
\][/tex]
3. Combine the simplified numerical part with the variable part:
So now we have:
[tex]\[
36 \cdot \frac{x^8}{x^2}
\][/tex]
4. Simplify the variable part:
Subtract the exponents (using [tex]\(a^m / a^n = a^{m-n}\)[/tex]):
[tex]\[
\frac{x^8}{x^2} = x^{8-2} = x^6
\][/tex]
5. Combine this with the numerical part:
This gives us:
[tex]\[
36 x^6
\][/tex]
6. Now, we need to take the square root of the entire expression:
[tex]\[
\sqrt{36 x^6}
\][/tex]
7. Simplify the square root of the numerical part:
[tex]\[
\sqrt{36} = 6
\][/tex]
8. Simplify the square root of the variable part:
[tex]\[
\sqrt{x^6} = x^{6/2} = x^3
\][/tex]
9. Combine these results:
[tex]\[
6 x^3
\][/tex]
So, the simplified form of [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex] is:
[tex]\[
6 x^3
\][/tex]
Thus, the answer is [tex]\(6 x^3\)[/tex], which matches one of the given options.
