Let's break down the problem into steps to obtain the solution:
1. Prime Factorization of the Radicand:
The radicand given is 135. First, we need to find its prime factors.
- 135 can be divided by 3:
- 135 ÷ 3 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
So the prime factorization of 135 is:
[tex]\[
135 = 3^3 \times 5
\][/tex]
2. Simplified Form of [tex]\(3 \sqrt{135}\)[/tex]:
Using the prime factorization, we can simplify [tex]\( \sqrt{135} \)[/tex]:
[tex]\[
\sqrt{135} = \sqrt{3^3 \times 5}
\][/tex]
We know that:
[tex]\[
\sqrt{3^3 \times 5} = \sqrt{3^2 \times 3 \times 5} = \sqrt{(3^2) \times (3 \times 5)}
\][/tex]
[tex]\[
\sqrt{(3^2) \times (3 \times 5)} = \sqrt{3^2} \times \sqrt{3 \times 5}
\][/tex]
[tex]\[
\sqrt{3^2} \times \sqrt{3 \times 5} = 3 \sqrt{15}
\][/tex]
Therefore:
[tex]\[
3 \sqrt{135} = 3 \times 3 \sqrt{15} = 9 \sqrt{15}
\][/tex]
3. Simplify [tex]\( 3(3) \sqrt{5(3)} \ = 9 \sqrt{15} \)[/tex]:
Let’s check the correctness:
- We formed [tex]\(\sqrt{15}\)[/tex] by recognizing [tex]\(15 = 3 \times 5\)[/tex].
- Our simplified form utilizes the product property of square roots to turn [tex]\(\sqrt{3^2 \times 5}\)[/tex] into [tex]\(3 \sqrt{15}\)[/tex].
Summing it all up, our previous steps lead to the simplified expression:
[tex]\[
3 \sqrt{135} = 9 \sqrt{15}
\][/tex]
