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1. Write the prime factorization of the radicand.

What is the simplified form of [tex]$3 \sqrt{135}$[/tex]?

2. Apply the product property of square roots. Write the radicand as a product, forming as many perfect square roots as possible.

[tex]$3 \sqrt{135} = 3 \sqrt{5 \cdot 3^3}$[/tex]

3. Simplify.

[tex]3 \sqrt{135} = 3 \cdot 3 \sqrt{15} = 9 \sqrt{15}$[/tex]


Sagot :

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]