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Sagot :
Certainly! Let's work through the expression [tex]\( \sqrt{20} \)[/tex] and simplify it to its simplest radical form step-by-step.
1. Identify the prime factors of the radicand:
- First, we need to factor the number under the square root. The number 20 can be factored into prime factors:
[tex]\[ 20 = 2 \times 10 \][/tex]
and then
[tex]\[ 10 = 2 \times 5 \][/tex]
So,
[tex]\[ 20 = 2 \times 2 \times 5 \][/tex]
2. Group the factors into pairs:
- The square root of a product can be broken down into the product of square roots. Let's group the pairs together:
[tex]\[ \sqrt{20} = \sqrt{2 \times 2 \times 5} \][/tex]
3. Simplify the square root:
- We can take the square root of each pair of factors. Since the square root of [tex]\(2 \times 2\)[/tex] is 2 (because [tex]\(\sqrt{2 \times 2} = 2\)[/tex]),
we can rewrite the expression as:
[tex]\[ \sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5} \][/tex]
Thus, the expression [tex]\( \sqrt{20} \)[/tex] in its simplest radical form is:
[tex]\[ 2\sqrt{5} \][/tex]
This is the simplified form of [tex]\( \sqrt{20} \)[/tex].
1. Identify the prime factors of the radicand:
- First, we need to factor the number under the square root. The number 20 can be factored into prime factors:
[tex]\[ 20 = 2 \times 10 \][/tex]
and then
[tex]\[ 10 = 2 \times 5 \][/tex]
So,
[tex]\[ 20 = 2 \times 2 \times 5 \][/tex]
2. Group the factors into pairs:
- The square root of a product can be broken down into the product of square roots. Let's group the pairs together:
[tex]\[ \sqrt{20} = \sqrt{2 \times 2 \times 5} \][/tex]
3. Simplify the square root:
- We can take the square root of each pair of factors. Since the square root of [tex]\(2 \times 2\)[/tex] is 2 (because [tex]\(\sqrt{2 \times 2} = 2\)[/tex]),
we can rewrite the expression as:
[tex]\[ \sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5} \][/tex]
Thus, the expression [tex]\( \sqrt{20} \)[/tex] in its simplest radical form is:
[tex]\[ 2\sqrt{5} \][/tex]
This is the simplified form of [tex]\( \sqrt{20} \)[/tex].
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