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].
