To determine which expression is in its simplest form, we need to simplify each expression and compare them.
1. Simplify [tex]\(3 \sqrt{8}\)[/tex]:
- Note that [tex]\(\sqrt{8}\)[/tex] can be rewritten using its factors: [tex]\(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\)[/tex].
- Now substitute back: [tex]\(3 \sqrt{8} = 3 \times 2 \sqrt{2} = 6 \sqrt{2}\)[/tex].
2. Simplify [tex]\(\sqrt{132}\)[/tex]:
- Note that [tex]\(\sqrt{132}\)[/tex] can be rewritten using its factors: [tex]\(\sqrt{132} = \sqrt{4 \times 33} = \sqrt{4} \times \sqrt{33} = 2 \sqrt{33}\)[/tex].
- The simplified form is [tex]\(2 \sqrt{33}\)[/tex].
3. Simplify [tex]\(\sqrt{142}\)[/tex]:
- [tex]\(\sqrt{142}\)[/tex] cannot be simplified further since 142 is not a product of a perfect square and another number. Therefore, the simplest form is [tex]\(\sqrt{142}\)[/tex].
4. Simplify [tex]\(2 \sqrt{63}\)[/tex]:
- Note that [tex]\(\sqrt{63}\)[/tex] can be rewritten using its factors: [tex]\(\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3 \sqrt{7}\)[/tex].
- Now substitute back: [tex]\(2 \sqrt{63} = 2 \times 3 \sqrt{7} = 6 \sqrt{7}\)[/tex].
So, the simplified expressions are:
- [tex]\(6 \sqrt{2}\)[/tex]
- [tex]\(2 \sqrt{33}\)[/tex]
- [tex]\(\sqrt{142}\)[/tex]
- [tex]\(6 \sqrt{7}\)[/tex]
Among these expressions, the simplest forms are [tex]\(\sqrt{132}\)[/tex] and [tex]\(\sqrt{142}\)[/tex]. Since both are in simplest form already but [tex]\(\sqrt{132}\)[/tex] when simplified becomes [tex]\(2\sqrt{33}\)[/tex] is the correct value. Therefore, the correct expression in its simplest form is:
[tex]\(\boxed{\sqrt{132}}\)[/tex]
