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Simplify the following expression: [tex]\sqrt{108}[/tex]

Sagot :

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]

Answer:

6√3

Step-by-step explanation:

To simplify √108:

Find the factors of 108:

108 = 36 × 3

Simplify:

√108

√( 36 × 3)

√36 × √3

6√3

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