To simplify the expression [tex]\(\sqrt{\frac{27 x^{12}}{300 x^8}}\)[/tex], let's go through the steps step-by-step.
1. Simplify the Fraction inside the Square Root:
[tex]\[
\frac{27 x^{12}}{300 x^8}
\][/tex]
2. Separate the Numerical and Variable Parts:
[tex]\[
\sqrt{\frac{27}{300} \cdot \frac{x^{12}}{x^8}}
\][/tex]
3. Simplify the Numerical Fraction:
[tex]\[
\frac{27}{300}
\][/tex]
We can simplify [tex]\(\frac{27}{300}\)[/tex]:
[tex]\[
\frac{27}{300} = \frac{27 \div 3}{300 \div 3} = \frac{9}{100}
\][/tex]
So, the expression becomes:
[tex]\[
\sqrt{\frac{9}{100} \cdot \frac{x^{12}}{x^8}}
\][/tex]
4. Simplify the Variable Part:
[tex]\[
\frac{x^{12}}{x^8} = x^{12-8} = x^4
\][/tex]
Now the expression is:
[tex]\[
\sqrt{\frac{9}{100} \cdot x^4}
\][/tex]
5. Combine and Simplify under the Square Root:
[tex]\[
\sqrt{\frac{9 x^4}{100}}
\][/tex]
6. Distribute the Square Root to Numerator and Denominator:
[tex]\[
\frac{\sqrt{9 x^4}}{\sqrt{100}}
\][/tex]
Simplify the components:
[tex]\[
\sqrt{9 x^4} = 3 x^2
\][/tex]
[tex]\[
\sqrt{100} = 10
\][/tex]
Therefore, combining these results:
[tex]\[
\frac{3 x^2}{10}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{3}{10} x^2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{3}{10} x^2}
\][/tex]
