Discover a wealth of information and get your questions answered on IDNLearn.com. Ask any question and get a thorough, accurate answer from our community of experienced professionals.
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]
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.