Sure! Let’s simplify the expression [tex]\(\sqrt{27 u^{11}}\)[/tex] step-by-step.
### Step 1: Factorize the expression inside the square root
First, we'll break down the number 27 and the term [tex]\(u^{11}\)[/tex]:
[tex]\[
27 = 3^3
\][/tex]
Thus, the expression can be written as:
[tex]\[
\sqrt{27 u^{11}} = \sqrt{3^3 \cdot u^{11}}
\][/tex]
### Step 2: Separate the square root
We can use the property of square roots [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{3^3 \cdot u^{11}} = \sqrt{3^3} \cdot \sqrt{u^{11}}
\][/tex]
### Step 3: Simplify each part separately
Next, we evaluate each square root separately.
- Simplify [tex]\(\sqrt{3^3}\)[/tex]:
[tex]\[
\sqrt{3^3} = \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\][/tex]
- Simplify [tex]\(\sqrt{u^{11}}\)[/tex]:
Since [tex]\(u^{11} = (u^{10} \cdot u)\)[/tex] and we know the square root of [tex]\(u^{10}\)[/tex] (as it is even), we get:
[tex]\[
\sqrt{u^{11}} = \sqrt{u^{10} \cdot u} = \sqrt{(u^5)^2 \cdot u} = u^5 \cdot \sqrt{u}
\][/tex]
Combining both parts, we get:
[tex]\[
\sqrt{3^3} \cdot \sqrt{u^{11}} = 3\sqrt{3} \cdot u^5 \cdot \sqrt{u}
\][/tex]
### Step 4: Combine the simplified parts
Finally, we combine everything to get the simplified form of the original expression:
[tex]\[
\sqrt{27 u^{11}} = 3\sqrt{3} \cdot u^5 \cdot \sqrt{u}
\][/tex]
We can also write it as:
[tex]\[
3\sqrt{3} \cdot \sqrt{u^{11}}
\][/tex]
So, combining it into one expression, the simplified form is:
[tex]\[
3\sqrt{3} \sqrt{u^{11}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{3\sqrt{3} \sqrt{u^{11}}}
\][/tex]
