Sure, let's simplify the expression [tex]\(\sqrt{27 u^{11}}\)[/tex] step by step.
1. Express the Radicand as a Product of Square and Non-Square Factors:
We can begin by breaking down [tex]\(27\)[/tex] and [tex]\(u^{11}\)[/tex] into factors that include perfect squares.
Note that [tex]\(27\)[/tex] can be written as [tex]\(3^3\)[/tex] and [tex]\(u^{11}\)[/tex] is already in exponential form with the variable [tex]\(u\)[/tex].
So, we have:
[tex]\[
27 u^{11} = 3^3 u^{11}
\][/tex]
2. Rewrite the Expression:
We want to separate the expression into parts that can include perfect squares:
[tex]\[
\sqrt{27 u^{11}} = \sqrt{3^3 \cdot u^{11}}
\][/tex]
3. Apply the Property of Square Root to the Product:
Use the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex] to split the expression:
[tex]\[
\sqrt{3^3 \cdot u^{11}} = \sqrt{3^3} \cdot \sqrt{u^{11}}
\][/tex]
4. Simplify Each Part Separately:
- For [tex]\(\sqrt{3^3}\)[/tex]:
Since [tex]\(3^3 = 3 \cdot 3 \cdot 3 = 9 \cdot 3\)[/tex], we can take the square root of [tex]\(9\)[/tex]:
[tex]\[
\sqrt{3^3} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \cdot \sqrt{3}
\][/tex]
- For [tex]\(\sqrt{u^{11}}\)[/tex]:
We know that [tex]\(u^{11} = (u^5)^2 \cdot u\)[/tex]:
[tex]\[
\sqrt{u^{11}} = \sqrt{(u^5)^2 \cdot u} = u^5 \cdot \sqrt{u}
\][/tex]
5. Combine the Parts Together:
Now, combine the simplified parts together:
[tex]\[
\sqrt{27 u^{11}} = 3 \cdot \sqrt{3} \cdot u^5 \cdot \sqrt{u}
\][/tex]
6. Simplify Further if Possible:
Combine the square root parts:
[tex]\[
3 \cdot \sqrt{3} \cdot u^5 \cdot \sqrt{u} = 3 \cdot \sqrt{3} \cdot u^5 \cdot \sqrt{u}
\][/tex]
So, the simplified form of [tex]\(\sqrt{27 u^{11}}\)[/tex] is:
[tex]\[
3\sqrt{3} \sqrt{u^{11}}
\][/tex]
or equivalently:
[tex]\[
3\sqrt{3} \sqrt{u^5 \cdot u} = 3\sqrt{3} \sqrt{u^5} \sqrt{u} = 3\sqrt{3} u^5 \sqrt{u}
\][/tex]
Therefore, the final simplified result is:
[tex]\[
3\sqrt{3} \sqrt{u^{11}}
\][/tex]
