Certainly! Let's simplify the given expression and compare it to each of the provided options.
Given Expression:
[tex]\[ \sqrt{80} \][/tex]
First, simplify the given expression [tex]\(\sqrt{80}\)[/tex].
[tex]\[ \sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \cdot \sqrt{5} \][/tex]
So, the simplified form of the given expression is:
[tex]\[ \sqrt{80} = 4 \sqrt{5} \][/tex]
Now, let's compare this to the provided options:
1. Option 1: [tex]\(80^{\frac{1}{2}}\)[/tex]
[tex]\[ 80^{\frac{1}{2}} = \sqrt{80} \][/tex]
As noted before, [tex]\(\sqrt{80} = 4 \sqrt{5}\)[/tex]. Thus, [tex]\(80^{\frac{1}{2}}\)[/tex] simplifies to [tex]\(4 \sqrt{5}\)[/tex].
2. Option 2: [tex]\(8 \sqrt{5}\)[/tex]
This expression clearly is not simplified to the same form as [tex]\(4 \sqrt{5}\)[/tex].
3. Option 3: [tex]\(4 \sqrt{10}\)[/tex]
This expression is in a different form and cannot be simplified to match [tex]\(4 \sqrt{5}\)[/tex].
4. Option 4: [tex]\(4 \sqrt{5}\)[/tex]
This expression directly matches [tex]\(4 \sqrt{5}\)[/tex].
5. Option 5: [tex]\(160^{\frac{1}{2}}\)[/tex]
[tex]\[ 160^{\frac{1}{2}} = \sqrt{160} \][/tex]
But [tex]\(\sqrt{160} = \sqrt{16 \cdot 10} = \sqrt{16} \cdot \sqrt{10} = 4 \cdot \sqrt{10} \)[/tex], which does not match [tex]\(4 \sqrt{5}\)[/tex].
Conclusion:
The expressions equivalent to [tex]\(\sqrt{80}\)[/tex] are:
[tex]\[ \boxed{80^{\frac{1}{2}}, 4 \sqrt{5}} \][/tex]
Hence, the correct options are:
[tex]\[ \boxed{80^{\frac{1}{2}}, 4 \sqrt{5}} \][/tex]
Therefore, the selected correct answers are:
- [tex]\( 80^{\frac{1}{2}} \)[/tex]
- [tex]\( 4 \sqrt{5} \)[/tex]
