To simplify the given 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]
2. Factor out the constants and powers of [tex]\(x\)[/tex]:
[tex]\[
= \frac{2160}{60} \cdot \frac{x^8}{x^2}
\][/tex]
3. Simplify the fraction with constants:
[tex]\[
2160 \div 60 = 36
\][/tex]
So, the expression now is:
[tex]\[
36 \cdot \frac{x^8}{x^2}
\][/tex]
4. Simplify the powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{x^8}{x^2} = x^{8-2} = x^6
\][/tex]
So, the expression now is:
[tex]\[
36 x^6
\][/tex]
5. Take the square root of the simplified expression:
[tex]\[
\sqrt{36 x^6}
\][/tex]
6. Simplify the square root:
Remember that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
Hence:
[tex]\[
\sqrt{36 x^6} = \sqrt{36} \cdot \sqrt{x^6}
\][/tex]
7. Evaluate the square roots:
[tex]\[
\sqrt{36} = 6
\][/tex]
For the powers of [tex]\(x\)[/tex]:
[tex]\[
\sqrt{x^6} = x^{6/2} = x^3
\][/tex]
8. Combine the results:
[tex]\[
6 \cdot x^3 = 6 x^3
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{\frac{2160 x^8}{60 x^2}}\)[/tex] is [tex]\(6 x^3\)[/tex].
The correct answer is:
[tex]\[
6 x^3
\][/tex]
