To simplify the given expression [tex]\(\sqrt{49 z^{10}}\)[/tex], follow these steps:
1. Understand the expression: The expression inside the square root is [tex]\(49 z^{10}\)[/tex].
2. Factor the radicand (the expression inside the square root): Notice that 49 is a perfect square. We can rewrite 49 as [tex]\(7^2\)[/tex].
[tex]\[
\sqrt{49 z^{10}} = \sqrt{(7^2) z^{10}}
\][/tex]
3. Apply the property of square roots: The square root of a product is the product of the square roots:
[tex]\[
\sqrt{(7^2) z^{10}} = \sqrt{(7^2)} \cdot \sqrt{z^{10}}
\][/tex]
4. Simplify the square root of a perfect square: The square root of [tex]\(7^2\)[/tex] is simply 7:
[tex]\[
\sqrt{(7^2)} = 7
\][/tex]
5. Simplify the square root of a power of [tex]\(z\)[/tex]: For the term [tex]\(z^{10}\)[/tex], we can use the rule [tex]\(\sqrt{a^2} = a\)[/tex]. Since [tex]\(10\)[/tex] is an even number, we can write [tex]\(z^{10}\)[/tex] as [tex]\((z^5)^2\)[/tex]:
[tex]\[
\sqrt{z^{10}} = \sqrt{(z^5)^2} = z^5
\][/tex]
6. Combine the simplified parts: Putting it all together, we have:
[tex]\[
\sqrt{49 z^{10}} = 7 \sqrt{z^{10}} = 7 \cdot z^5
\][/tex]
However, we can note that simplification of the square root of [tex]\(z^{10}\)[/tex] might be interpreted supifically as:
[tex]\[
\sqrt{z^10} = \sqrt{(z^5)^2} = z^5
\][/tex]
This confirms that the simplified form of [tex]\(\sqrt{49 z^{10}}\)[/tex] results in:
[tex]\[
7 \cdot z^5 = 7z^5
\][/tex]
Therefore, the final simplified expression is:
[tex]\[
7 \sqrt{z^{10}}
\][/tex]
