To simplify [tex]\(\sqrt{49z^4}\)[/tex], let's take it step by step.
1. Understand the Given Expression:
We have [tex]\(\sqrt{49z^4}\)[/tex], which can be written as:
[tex]\[
\sqrt{(49z^4)}
\][/tex]
2. Apply the Square Root to Each Component:
The expression inside the square root can be broken down into the product of two parts:
[tex]\[
\sqrt{49 \cdot z^4}
\][/tex]
Using the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex], we can rewrite it as:
[tex]\[
\sqrt{49} \cdot \sqrt{z^4}
\][/tex]
3. Simplify Each Part Separately:
- [tex]\(\sqrt{49}\)[/tex]: The square root of 49 is 7, since [tex]\(7 \times 7 = 49\)[/tex]:
[tex]\[
\sqrt{49} = 7
\][/tex]
- [tex]\(\sqrt{z^4}\)[/tex]: The square root of [tex]\(z^4\)[/tex] is [tex]\(z^2\)[/tex], because:
[tex]\[
(z^2)^2 = z^4
\][/tex]
Thus:
[tex]\[
\sqrt{z^4} = z^2
\][/tex]
4. Combine the Results:
Now combine the simplified parts:
[tex]\[
\sqrt{49} \cdot \sqrt{z^4} = 7 \cdot z^2 = 7z^2
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{49z^4}\)[/tex] is:
[tex]\[
7z^2
\][/tex]
Upon reviewing the provided options:
- Option [tex]\(a\)[/tex] is [tex]\(7z\)[/tex]
- Option [tex]\(b\)[/tex] is [tex]\(14z\)[/tex]
- Option [tex]\(c\)[/tex] is [tex]\(21z\)[/tex]
- Option [tex]\(d\)[/tex] is [tex]\(28z\)[/tex]
None of these options match [tex]\(7z^2\)[/tex]. There appears to be a mistake in the provided choices. The correct simplified expression is [tex]\(7z^2\)[/tex].
