To simplify the expression [tex]\(\sqrt{p^4 q^6 r^5}\)[/tex], we can follow these steps:
1. Write the given expression:
[tex]\[\sqrt{p^4 q^6 r^5}\][/tex]
2. Simplify the expression inside the square root:
Each of the variables is raised to a power, and we can decompose these powers to simplified bases.
3. Break down the expression inside the square root:
The expression can be broken down into:
[tex]\[
\sqrt{(p^4) \cdot (q^6) \cdot (r^4 \cdot r)}
\][/tex]
4. Simplify each term individually:
The square root of any variable to an even power is that variable to half the power:
[tex]\[\sqrt{p^4} = p^{4/2} = p^2\][/tex]
[tex]\[\sqrt{q^6} = q^{6/2} = q^3\][/tex]
[tex]\[\sqrt{r^4} = r^{4/2} = r^2\][/tex]
Since [tex]\(r^5\)[/tex] can be written as [tex]\(r^4 \cdot r\)[/tex], we have already simplified [tex]\(r^4\)[/tex] but we still have [tex]\(\sqrt{r}\)[/tex] left under the square root.
5. Combine the simplified components:
[tex]\[
\sqrt{p^4 q^6 r^5} = p^2 \cdot q^3 \cdot r^2 \cdot \sqrt{r}
\][/tex]
So the simplified form of the given expression is:
[tex]\[
p^2 q^3 r^2 \sqrt{r}
\][/tex]
Now we compare the simplified form to the given options:
- (a) [tex]\(p^2 q^2 r \cdot \sqrt[\frac{1}{3}]{p r^2}\)[/tex] does not match.
- (c) [tex]\(p q^2 r \cdot \sqrt[2]{p r^5}\)[/tex] does not match.
- (b) [tex]\(p^2 q^3 r \cdot \sqrt[3]{p r^2}\)[/tex] does not match.
- (d) [tex]\(p q^2 r \cdot \sqrt[3]{p r^2}\)[/tex] does not match.
None of the provided options match the correct simplified form [tex]\(p^2 q^3 r^2 \sqrt{r}\)[/tex].
Given that none of the options provided exactly match the simplified form we’ve calculated, it appears there might be an error in the options given.
However, based on our detailed and correct simplification, the correct form is:
[tex]\[
\boxed{p^2 q^3 r^2 \sqrt{r}}
\][/tex]
