To determine which expressions are equivalent to [tex]\(\sqrt{80}\)[/tex], we will analyze each one:
1. [tex]\(8 \sqrt{5}\)[/tex]:
- Let's simplify [tex]\(8 \sqrt{5}\)[/tex]:
[tex]\[
8 \sqrt{5} = 8 \times 2.2361 \approx 17.888
\][/tex]
- Now, let's compute [tex]\(\sqrt{80}\)[/tex]:
[tex]\[
\sqrt{80} \approx 8.944
\][/tex]
- Clearly, [tex]\(8 \sqrt{5}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
2. [tex]\(80^{\frac{1}{2}}\)[/tex]:
- By definition, [tex]\(\sqrt{80}\)[/tex] is equivalent to [tex]\(80^{\frac{1}{2}}\)[/tex]:
[tex]\[
\sqrt{80} = 80^{\frac{1}{2}} \approx 8.944
\][/tex]
- They are indeed equivalent.
3. [tex]\(4 \sqrt{5}\)[/tex]:
- Let's simplify [tex]\(4 \sqrt{5}\)[/tex]:
[tex]\[
4 \sqrt{5} = 4 \times 2.2361 \approx 8.944
\][/tex]
- The value matches [tex]\(\sqrt{80}\)[/tex], so [tex]\(4 \sqrt{5}\)[/tex] is equivalent to [tex]\(\sqrt{80}\)[/tex].
4. [tex]\(160^{\frac{1}{2}}\)[/tex]:
- Let's compute [tex]\(\sqrt{160}\)[/tex]:
[tex]\[
\sqrt{160} \approx 12.649
\][/tex]
- This does not match [tex]\(\sqrt{80}\)[/tex], so it is not equivalent.
5. [tex]\(4 \sqrt{10}\)[/tex]:
- Let's simplify [tex]\(4 \sqrt{10}\)[/tex]:
[tex]\[
4 \sqrt{10} = 4 \times 3.1623 \approx 12.649
\][/tex]
- This does not match [tex]\(\sqrt{80}\)[/tex], so it is not equivalent.
Based on the above analysis, the expressions that are equivalent to [tex]\(\sqrt{80}\)[/tex] are:
- [tex]\(80^{\frac{1}{2}}\)[/tex]
- [tex]\(4 \sqrt{5}\)[/tex]
Thus, the correct answers are:
[tex]\[ 80^{\frac{1}{2}} \quad \text{and} \quad 4 \sqrt{5} \][/tex]
