Alright, let’s work through the problem of expressing \(\sqrt{8}\) in its simplest radical form step by step.
1. Prime Factorization of 8:
- 8 can be broken down into its prime factors. We have:
[tex]\[
8 = 2 \times 4 = 2 \times 2 \times 2 = 2^3
\][/tex]
2. Simplifying the Radical:
- We want to separate the factors inside the square root into pairs of identical factors, which simplifies outside the square root.
- Recall that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Here, we are looking for pairs of identical factors under the square root since \(\sqrt{a^2} = a\).
3. Handling Perfect Squares:
- Since \(8 = 2 \times 2 \times 2\), we recognize that we can pair two of the 2's:
[tex]\[
\sqrt{8} = \sqrt{2 \times 2 \times 2}
\][/tex]
- We can take a pair of 2's out of the square root, as:
[tex]\[
\sqrt{2 \times 2 \times 2} = \sqrt{(2 \times 2) \times 2}
\][/tex]
- Thus, \(\sqrt{(2 \times 2) \times 2} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\).
4. Result:
- The simplest radical form of \(\sqrt{8}\) is \(2\sqrt{2}\).
So, [tex]\(\sqrt{8}\)[/tex] in its simplest radical form is [tex]\(2\sqrt{2}\)[/tex].
