Alright, let's simplify the given expression [tex]\(\sqrt{18 q^4}\)[/tex] step by step.
1. Step 1: Break down the expression inside the square root.
We have [tex]\(18 q^4\)[/tex]. Notice that this can be written as the product of two separate terms:
[tex]\[
18 q^4 = 18 \cdot q^4
\][/tex]
2. Step 2: Factorize the constant term.
We can break down [tex]\(18\)[/tex] into its prime factors:
[tex]\[
18 = 2 \cdot 3^2
\][/tex]
3. Step 3: Rewrite the expression using the factored form.
Substituting back into the original expression within the square root, we get:
[tex]\[
\sqrt{18 q^4} = \sqrt{2 \cdot 3^2 \cdot q^4}
\][/tex]
4. Step 4: Use the property of square roots that states [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)[/tex].
Applying this property, we can split the square root:
[tex]\[
\sqrt{2 \cdot 3^2 \cdot q^4} = \sqrt{2} \cdot \sqrt{3^2} \cdot \sqrt{q^4}
\][/tex]
5. Step 5: Simplify the square roots individually.
For [tex]\(\sqrt{3^2}\)[/tex]:
[tex]\[
\sqrt{3^2} = 3
\][/tex]
For [tex]\(\sqrt{q^4}\)[/tex]:
[tex]\[
\sqrt{q^4} = q^2
\][/tex]
The [tex]\(\sqrt{2}\)[/tex] term remains as it is since [tex]\(2\)[/tex] is not a perfect square.
6. Step 6: Combine the simplified terms.
Putting it all together, we get:
[tex]\[
\sqrt{2} \cdot 3 \cdot q^2
\][/tex]
7. Step 7: Arrange the terms in a simplified and conventional form.
The product of these terms can be written as:
[tex]\[
3 q^2 \sqrt{2}
\][/tex]
So, the simplified form of [tex]\(\sqrt{18 q^4}\)[/tex] is:
[tex]\[
3 q^2 \sqrt{2}
\][/tex]
Therefore, the final answer is:
[tex]\[
3 \sqrt{2} q^2
\][/tex]
