Certainly! Let's simplify the expression:
[tex]\[
\sqrt{25 z^{12}}
\][/tex]
### Step-by-Step Solution
1. Identify the components inside the square root:
[tex]\[
\sqrt{25 z^{12}}
\][/tex]
Here, we have the product of 25 and [tex]\(z^{12}\)[/tex].
2. Simplify the square root of a product:
Recall that the square root of a product can be split into the product of the square roots:
[tex]\[
\sqrt{25 z^{12}} = \sqrt{25} \cdot \sqrt{z^{12}}
\][/tex]
3. Evaluate the square root of 25:
[tex]\[
\sqrt{25} = 5
\][/tex]
4. Evaluate the square root of [tex]\(z^{12}\)[/tex]:
When taking the square root of a variable raised to an even power, we can halve the exponent:
[tex]\[
\sqrt{z^{12}} = z^{\frac{12}{2}} = z^6
\][/tex]
5. Combine the results:
We now combine the simplified components:
[tex]\[
\sqrt{25 z^{12}} = 5 \cdot z^6 = 5z^6
\][/tex]
### Final Simplified Expression
[tex]\[
\sqrt{25 z^{12}} = 5z^6
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt{25 z^{12}}\)[/tex] is [tex]\(\boxed{5z^6}\)[/tex].
