To simplify the expression [tex]\( \sqrt{12u^6} \)[/tex], we can follow a systematic approach to break it down into simpler parts:
1. Factor the expression under the square root.
2. Simplify each factor separately.
### Step-by-Step Solution:
1. Factor the expression under the square root:
Let's start by factoring [tex]\( 12u^6 \)[/tex].
[tex]\[
12 = 4 \times 3
\][/tex]
So, we can write:
[tex]\[
12u^6 = 4 \times 3 \times u^6
\][/tex]
2. Rewrite the square root expression using the above factors:
[tex]\[
\sqrt{12u^6} = \sqrt{4 \times 3 \times u^6}
\][/tex]
3. Apply the property of square roots to separate the factors:
The property of square roots states that [tex]\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)[/tex]. Using this property, we can rewrite our expression:
[tex]\[
\sqrt{4 \times 3 \times u^6} = \sqrt{4} \times \sqrt{3} \times \sqrt{u^6}
\][/tex]
4. Simplify each square root separately:
- [tex]\( \sqrt{4} = 2 \)[/tex], because 2 squared is 4.
- [tex]\( \sqrt{3} \)[/tex] remains [tex]\( \sqrt{3} \)[/tex].
- [tex]\( \sqrt{u^6} \)[/tex] can be simplified using the property that [tex]\( \sqrt{a^2} = a \)[/tex]:
[tex]\[
\sqrt{u^6} = (u^6)^{1/2} = u^{6 \cdot (1/2)} = u^3
\][/tex]
5. Combine the simplified parts:
[tex]\[
\sqrt{12u^6} = 2 \times \sqrt{3} \times u^3
\][/tex]
6. Write the final simplified expression:
[tex]\[
\sqrt{12u^6} = 2u^3 \sqrt{3}
\][/tex]
Therefore, the simplified form of [tex]\( \sqrt{12u^6} \)[/tex] is [tex]\( 2u^3 \sqrt{3} \)[/tex].
