To simplify [tex]\(\sqrt{108}\)[/tex], we need to break it down into its prime factors and identify perfect squares that can be factored out. Here is a step-by-step solution:
1. Prime Factorization: First, let's find the prime factors of 108.
[tex]\[
108 = 2 \times 54 = 2 \times (2 \times 27) = 2 \times 2 \times (3 \times 9) = 2 \times 2 \times 3 \times (3 \times 3)
\][/tex]
So,
[tex]\[
108 = 2^2 \times 3^3
\][/tex]
2. Rewrite the Square Root: Using the prime factorization, we can rewrite [tex]\(\sqrt{108}\)[/tex] as:
[tex]\[
\sqrt{108} = \sqrt{2^2 \times 3^3}
\][/tex]
3. Extract Perfect Squares: Identify the perfect squares within the factorization and extract them from under the square root.
[tex]\[
\sqrt{108} = \sqrt{2^2 \times 3^2 \times 3}
\][/tex]
[tex]\[
= \sqrt{(2^2) \times (3^2) \times 3}
\][/tex]
4. Simplify: Take the square roots of the perfect squares [tex]\(2^2\)[/tex] and [tex]\(3^2\)[/tex].
[tex]\[
\sqrt{108} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{3}
\][/tex]
[tex]\[
= 2 \times 3 \times \sqrt{3}
\][/tex]
[tex]\[
= 6\sqrt{3}
\][/tex]
Hence, the simplest radical form (R.F) of [tex]\(\sqrt{108}\)[/tex] is:
[tex]\[
6\sqrt{3}
\][/tex]
