To simplify the expression [tex]\(\sqrt{\frac{72 x^{16}}{50 x^{36}}}\)[/tex], let's break it down step-by-step:
1. Simplify the fraction inside the square root:
[tex]\[
\frac{72 x^{16}}{50 x^{36}}
\][/tex]
We can rewrite this as:
[tex]\[
\frac{72}{50} \cdot \frac{x^{16}}{x^{36}}
\][/tex]
2. Simplify the constants:
[tex]\[
\frac{72}{50}
\][/tex]
Simplify [tex]\(\frac{72}{50}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives:
[tex]\[
\frac{72}{50} = \frac{36}{25}
\][/tex]
3. Simplify the variable terms:
[tex]\[
\frac{x^{16}}{x^{36}}
\][/tex]
Using the properties of exponents, [tex]\(\frac{x^a}{x^b} = x^{a - b}\)[/tex]:
[tex]\[
\frac{x^{16}}{x^{36}} = x^{16 - 36} = x^{-20}
\][/tex]
4. Combine the simplified constants and variable terms:
The expression now becomes:
[tex]\[
\sqrt{\frac{36}{25} \cdot x^{-20}}
\][/tex]
5. Distribute the square root:
[tex]\[
\sqrt{\frac{36}{25} \cdot x^{-20}} = \sqrt{\frac{36}{25}} \cdot \sqrt{x^{-20}}
\][/tex]
6. Simplify each square root individually:
[tex]\[
\sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5}
\][/tex]
For the variable term:
[tex]\[
\sqrt{x^{-20}} = (x^{-20})^{1/2} = x^{-10}
\][/tex]
7. Combine the results:
The simplified form is:
[tex]\[
\frac{6}{5} \cdot x^{-10} = \frac{6}{5x^{10}}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{\frac{72 x^{16}}{50 x^{36}}}\)[/tex] is:
[tex]\[
\frac{6}{5x^{10}}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{6}{5x^{10}}}
\][/tex]
