Let's examine each of the given expressions to determine which ones are equivalent to [tex]\(\sqrt{40}\)[/tex].
1. [tex]\(40^{\frac{1}{2}}\)[/tex]:
- The notation [tex]\(40^{\frac{1}{2}}\)[/tex] represents the square root of 40.
- Thus, [tex]\(40^{\frac{1}{2}} = \sqrt{40}\)[/tex].
- Therefore, [tex]\(\sqrt{40}\)[/tex] is equivalent to [tex]\(40^{\frac{1}{2}}\)[/tex].
2. [tex]\(4 \sqrt{10}\)[/tex]:
- Let's break down [tex]\(4 \sqrt{10}\)[/tex].
- [tex]\(\sqrt{40}\)[/tex] can be factored as follows: [tex]\(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10}\)[/tex].
- Clearly, [tex]\(4 \sqrt{10}\)[/tex] is not equivalent to [tex]\(2 \sqrt{10}\)[/tex].
3. [tex]\(5 \sqrt{8}\)[/tex]:
- Let's break down [tex]\(5 \sqrt{8}\)[/tex].
- [tex]\(\sqrt{8}\)[/tex] can be rewritten as [tex]\(\sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\)[/tex].
- So, [tex]\(5 \sqrt{8} = 5 \times 2 \sqrt{2} = 10 \sqrt{2}\)[/tex].
- Clearly, [tex]\(\sqrt{40} \neq 10 \sqrt{2}\)[/tex].
4. [tex]\(160^{\frac{1}{2}}\)[/tex]:
- The notation [tex]\(160^{\frac{1}{2}}\)[/tex] represents the square root of 160.
- Let's compare [tex]\(\sqrt{160}\)[/tex] to [tex]\(\sqrt{40}\)[/tex].
- Clearly, [tex]\(\sqrt{40} \neq \sqrt{160}\)[/tex].
5. [tex]\(2 \sqrt{10}\)[/tex]:
- From the earlier factorization, we have seen that [tex]\(\sqrt{40} = 2 \sqrt{10}\)[/tex].
- Therefore, [tex]\(\sqrt{40}\)[/tex] is equivalent to [tex]\(2 \sqrt{10}\)[/tex].
In summary, the expressions equivalent to [tex]\(\sqrt{40}\)[/tex] are:
- [tex]\( 40^{\frac{1}{2}} \)[/tex]
- [tex]\( 2 \sqrt{10} \)[/tex]
So the correct answers are:
- [tex]\( 40^{\frac{1}{2}} \)[/tex]
- [tex]\( 2 \sqrt{10} \)[/tex]
