To determine which expressions are equivalent to [tex]\(\sqrt{40}\)[/tex], we need to simplify and compare each option step-by-step.
### Given Expression:
[tex]\(\sqrt{40}\)[/tex]
### Option 1: [tex]\(40^{\frac{1}{2}}\)[/tex]
This expression represents the same operation as [tex]\(\sqrt{40}\)[/tex].
[tex]\[
40^{\frac{1}{2}} = \sqrt{40}
\][/tex]
So, [tex]\(40^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex].
### Option 2: [tex]\(160^{\frac{1}{2}}\)[/tex]
This expression represents the square root of 160.
[tex]\[
160^{\frac{1}{2}} = \sqrt{160}
\][/tex]
Since [tex]\(\sqrt{160}\)[/tex] is not equal to [tex]\(\sqrt{40}\)[/tex], [tex]\(160^{\frac{1}{2}}\)[/tex] is not equivalent to [tex]\(\sqrt{40}\)[/tex].
### Option 3: [tex]\(5 \sqrt{8}\)[/tex]
To compare this, let's simplify it:
[tex]\[
5 \sqrt{8} = 5 \sqrt{4 \cdot 2} = 5 \cdot 2 \sqrt{2} = 10 \sqrt{2}
\][/tex]
To see if it is equivalent, let's simplify [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
\sqrt{40} = \sqrt{4 \cdot 10} = 2 \sqrt{10}
\][/tex]
Since [tex]\(\sqrt{40} \neq 10 \sqrt{2}\)[/tex], [tex]\(5 \sqrt{8}\)[/tex] is not equivalent to [tex]\(\sqrt{40}\)[/tex].
### Option 4: [tex]\(2 \sqrt{10}\)[/tex]
Simplify:
[tex]\[
2 \sqrt{10}
\][/tex]
We previously simplified [tex]\(\sqrt{40}\)[/tex] to [tex]\(2 \sqrt{10}\)[/tex], so:
[tex]\[
\sqrt{40} = 2 \sqrt{10}
\][/tex]
Therefore, [tex]\(2 \sqrt{10}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex].
### Option 5: [tex]\(4 \sqrt{10}\)[/tex]
Simplify:
[tex]\[
4 \sqrt{10}
\][/tex]
Since [tex]\(\sqrt{40} = 2 \sqrt{10}\)[/tex], and [tex]\(4 \sqrt{10}\)[/tex] is not equal to [tex]\(2 \sqrt{10}\)[/tex], [tex]\(4 \sqrt{10}\)[/tex] is not equivalent to [tex]\(\sqrt{40}\)[/tex].
### Summary:
The expressions that are equivalent to [tex]\(\sqrt{40}\)[/tex] are:
- [tex]\(40^{\frac{1}{2}}\)[/tex]
- [tex]\(2 \sqrt{10}\)[/tex]
Thus, in conclusion, the correct answers are:
- [tex]\(40^{\frac{1}{2}}\)[/tex]
- [tex]\(2 \sqrt{10}\)[/tex]
