To simplify [tex]\(\sqrt{32}\)[/tex], we need to break down the number 32 into its prime factors and then simplify the square root. Let's do it step-by-step:
1. Prime factorization of 32:
[tex]\[
32 = 2^5
\][/tex]
2. Rewrite the square root using the prime factorization:
[tex]\[
\sqrt{32} = \sqrt{2^5}
\][/tex]
3. Factor out squares:
We can rewrite [tex]\(\sqrt{2^5}\)[/tex] as [tex]\(\sqrt{(2^4) \cdot 2}\)[/tex]:
[tex]\[
\sqrt{2^5} = \sqrt{(2^4) \cdot 2} = \sqrt{2^4} \cdot \sqrt{2}
\][/tex]
4. Simplify [tex]\(\sqrt{2^4}\)[/tex]:
[tex]\[
\sqrt{2^4} = 2^2 = 4
\][/tex]
5. Combine the simplified terms:
[tex]\[
\sqrt{32} = 4 \cdot \sqrt{2}
\][/tex]
So, the simplified form of [tex]\(\sqrt{32}\)[/tex] is:
[tex]\[
\boxed{4 \sqrt{2}}
\][/tex]
