To simplify the expression [tex]\(\sqrt{\frac{72 x^{16}}{50 x^{36}}}\)[/tex], we need to handle both the numerical coefficients and the variable [tex]\(x\)[/tex] separately. Let's go through this process step-by-step.
1. Simplify the Coefficients:
[tex]\[
\sqrt{\frac{72}{50}}
\][/tex]
We can simplify [tex]\(\frac{72}{50}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{72}{50} = \frac{36}{25}
\][/tex]
So we need to find the square root of [tex]\(\frac{36}{25}\)[/tex]:
[tex]\[
\sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5}
\][/tex]
2. Simplify the Exponents of [tex]\(x\)[/tex]:
[tex]\[
\sqrt{\frac{x^{16}}{x^{36}}}
\][/tex]
When dividing exponents with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{x^{16}}{x^{36}} = x^{16 - 36} = x^{-20}
\][/tex]
Now we take the square root of [tex]\(x^{-20}\)[/tex]:
[tex]\[
\sqrt{x^{-20}} = x^{-20 \cdot \frac{1}{2}} = x^{-10}
\][/tex]
3. Combine the Results:
Putting the simplified coefficient and simplified exponent together, we get:
[tex]\[
\sqrt{\frac{72 x^{16}}{50 x^{36}}} = \frac{6}{5} x^{-10}
\][/tex]
In order form, [tex]\(\frac{6}{5} x^{-10}\)[/tex] can also be written as:
[tex]\[
\frac{6}{5 x^{10}}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{\frac{72 x^{16}}{50 x^{36}}}\)[/tex] is:
[tex]\[
\boxed{\frac{6}{5 x^{10}}}
\][/tex]
