To simplify the expression [tex]\(\sqrt{18 f^8}\)[/tex], we can follow these steps:
1. Prime Factorization: First, factorize the number inside the square root.
- The number 18 can be factorized as [tex]\(18 = 2 \times 3^2\)[/tex].
2. Rewrite the Expression: Next, substitute the factorized form back into the expression and break it down:
[tex]\[
\sqrt{18 f^8} = \sqrt{2 \cdot 3^2 \cdot f^8}
\][/tex]
3. Separate the Radicals: Since the square root of a product is the product of the square roots, separate the expression under the square root:
[tex]\[
\sqrt{2 \cdot 3^2 \cdot f^8} = \sqrt{2} \cdot \sqrt{3^2} \cdot \sqrt{f^8}
\][/tex]
4. Simplify Each Factor:
- The square root of [tex]\(3^2\)[/tex] is [tex]\(3\)[/tex]:
[tex]\[
\sqrt{3^2} = 3
\][/tex]
- The square root of [tex]\(f^8\)[/tex] can be simplified using the property [tex]\(\sqrt{(a^2)} = a\)[/tex]:
[tex]\[
\sqrt{f^8} = (f^8)^{1/2} = f^{8 \cdot \frac{1}{2}} = f^4
\][/tex]
5. Combine Results:
- Now, combine all the simplified parts:
[tex]\[
\sqrt{2} \cdot 3 \cdot f^4
\][/tex]
6. Reorder Factors: Arrange the factors in a conventional order:
[tex]\[
3 \cdot \sqrt{2} \cdot f^4
\][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{18 f^8}\)[/tex] is:
[tex]\[
3 \sqrt{2} f^4
\][/tex]
