To determine which expressions are equivalent to the given expression [tex]\(\sqrt{80}\)[/tex], let's analyze each expression step-by-step.
### Given expression:
[tex]\[
\sqrt{80}
\][/tex]
### Expressions to Compare:
1. [tex]\(160^{\frac{1}{2}}\)[/tex]
2. [tex]\(8 \sqrt{5}\)[/tex]
3. [tex]\(4 \sqrt{10}\)[/tex]
4. [tex]\(80^{\frac{1}{2}}\)[/tex]
5. [tex]\(4 \sqrt{5}\)[/tex]
### Step-by-Step Solution:
1. Expression [tex]\(160^{\frac{1}{2}}\)[/tex]:
- Taking the square root of 160:
[tex]\[
160^{\frac{1}{2}} = \sqrt{160}
\][/tex]
- [tex]\(\sqrt{160}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
2. Expression [tex]\(8 \sqrt{5}\)[/tex]:
- Rearranging the expression:
[tex]\[
8 \sqrt{5}
\][/tex]
- This expression is different from [tex]\(\sqrt{80}\)[/tex].
3. Expression [tex]\(4 \sqrt{10}\)[/tex]:
- Rearranging the expression:
[tex]\[
4 \sqrt{10}
\][/tex]
- This expression is different from [tex]\(\sqrt{80}\)[/tex].
4. Expression [tex]\(80^{\frac{1}{2}}\)[/tex]:
- Taking the exponent:
[tex]\[
80^{\frac{1}{2}} = \sqrt{80}
\][/tex]
- This expression is equivalent to [tex]\(\sqrt{80}\)[/tex].
5. Expression [tex]\(4 \sqrt{5}\)[/tex]:
- Rearranging the expression:
[tex]\[
4 \sqrt{5}
\][/tex]
- This expression is different from [tex]\(\sqrt{80}\)[/tex].
### Conclusion:
The expressions that are equivalent to [tex]\(\sqrt{80}\)[/tex] are:
[tex]\[
80^{\frac{1}{2}} \quad \text{and} \quad 4 \sqrt{5}
\][/tex]
