To simplify the expression [tex]\(\sqrt{54 v^{10}}\)[/tex], let's break it down step-by-step:
1. Factorize the inside expression:
[tex]\[
54 = 2 \times 3^3
\][/tex]
So, we can rewrite [tex]\(54\)[/tex] as:
[tex]\[
\sqrt{54 v^{10}} = \sqrt{2 \times 3^3 \times v^{10}}
\][/tex]
2. Separate the factors inside the square root:
According to the properties of square roots, we have:
[tex]\[
\sqrt{54 v^{10}} = \sqrt{2 \times 3^3 \times v^{10}} = \sqrt{2} \times \sqrt{3^3} \times \sqrt{v^{10}}
\][/tex]
3. Simplify each square root separately:
- [tex]\(\sqrt{2}\)[/tex]: This is already in its simplest form.
- [tex]\(\sqrt{3^3}\)[/tex]:
[tex]\[
3^3 = 27 \quad \Rightarrow \quad \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}
\][/tex]
- [tex]\(\sqrt{v^{10}}\)[/tex]:
[tex]\[
v^{10} = (v^5)^2 \quad \Rightarrow \quad \sqrt{v^{10}} = v^5
\][/tex]
4. Combine all the simplified parts:
Putting all the simplified parts together, we get:
[tex]\[
\sqrt{54 v^{10}} = \sqrt{2} \times 3\sqrt{3} \times v^5
\][/tex]
5. Simplify the expression:
Multiplying the constants and combining terms, we arrive at:
[tex]\[
\sqrt{54 v^{10}} = 3 \sqrt{2} \sqrt{3} v^5 = 3 \sqrt{2 \times 3} v^5 = 3 \sqrt{6} v^5
\][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{54 v^{10}}\)[/tex] is:
[tex]\[
3 \sqrt{6} v^5
\][/tex]
