To simplify the expression [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex], follow these steps:
1. Simplify the fraction inside the square root:
[tex]\[
\frac{2160 x^8}{60 x^2}
\][/tex]
Divide the coefficients:
[tex]\[
\frac{2160}{60} = 36
\][/tex]
Simplify the exponents of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^8}{x^2} = x^{8-2} = x^6
\][/tex]
So the expression simplifies to:
[tex]\[
36 x^6
\][/tex]
2. Take the square root of the simplified expression:
[tex]\[
\sqrt{36 x^6}
\][/tex]
3. Separate the square root into the product of two square roots:
[tex]\[
\sqrt{36} \cdot \sqrt{x^6}
\][/tex]
4. Calculate the square root of the constant term:
[tex]\[
\sqrt{36} = 6
\][/tex]
5. Simplify the square root of the variable term:
[tex]\[
\sqrt{x^6} = x^{6/2} = x^3
\][/tex]
6. Combine the results:
[tex]\[
6 \cdot x^3 = 6 x^3
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex] is:
[tex]\[
6 x^3
\][/tex]
So, the correct answer is:
[tex]\(6 x^3\)[/tex]
