Select all the correct answers.

Which expressions are equivalent to the given expression?

[tex]\[ \sqrt{80} \][/tex]

A. [tex]\( 8 \sqrt{5} \)[/tex]

B. [tex]\( 80^{\frac{1}{2}} \)[/tex]

C. [tex]\( 4 \sqrt{5} \)[/tex]

D. [tex]\( 160^{\frac{1}{2}} \)[/tex]

E. [tex]\( 4 \sqrt{10} \)[/tex]



Answer :

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]