To determine which expressions are equivalent to [tex]\(\sqrt{80}\)[/tex], let's evaluate each given option and see if they match [tex]\(\sqrt{80}\)[/tex].
1. Expression: [tex]\(80^{\frac{1}{2}}\)[/tex]
- This expression represents the square root of 80, written differently. So, [tex]\(80^{\frac{1}{2}} = \sqrt{80}\)[/tex].
- Therefore, [tex]\(80^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(\sqrt{80}\)[/tex].
2. Expression: [tex]\(4 \sqrt{5}\)[/tex]
- We need to check if [tex]\(4 \sqrt{5}\)[/tex] simplifies to [tex]\(\sqrt{80}\)[/tex].
- The expression [tex]\(4 \sqrt{5}\)[/tex] can be broken down as [tex]\(4 \times \sqrt{5}\)[/tex]. By squaring 4, we get 16, and multiplying that by 5 gives us [tex]\(16 \times 5 = 80\)[/tex]. So, [tex]\(4 \sqrt{5} = \sqrt{80}\)[/tex].
- Therefore, [tex]\(4 \sqrt{5}\)[/tex] is equivalent to [tex]\(\sqrt{80}\)[/tex].
3. Expression: [tex]\(8 \sqrt{5}\)[/tex]
- We need to check if [tex]\(8 \sqrt{5}\)[/tex] simplifies to [tex]\(\sqrt{80}\)[/tex].
- The expression [tex]\(8 \sqrt{5}\)[/tex] can be broken down as [tex]\(8 \times \sqrt{5}\)[/tex]. By squaring 8, we get 64, and multiplying that by 5 gives us [tex]\(64 \times 5 = 320\)[/tex]. So, [tex]\(8 \sqrt{5}\)[/tex] is not equal to [tex]\(\sqrt{80}\)[/tex].
- Therefore, [tex]\(8 \sqrt{5}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
4. Expression: [tex]\(160^{\frac{1}{2}}\)[/tex]
- This expression represents the square root of 160.
- Since [tex]\(\sqrt{160}\)[/tex] is not equal to [tex]\(\sqrt{80}\)[/tex], [tex]\(160^{\frac{1}{2}}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
5. Expression: [tex]\(4 \sqrt{10}\)[/tex]
- We need to check if [tex]\(4 \sqrt{10}\)[/tex] simplifies to [tex]\(\sqrt{80}\)[/tex].
- The expression [tex]\(4 \sqrt{10}\)[/tex] can be broken down as [tex]\(4 \times \sqrt{10}\)[/tex]. By squaring 4, we get 16, and multiplying that by 10 gives us [tex]\(16 \times 10 = 160\)[/tex]. So, [tex]\(4 \sqrt{10}\)[/tex] is not equal to [tex]\(\sqrt{80}\)[/tex].
- Therefore, [tex]\(4 \sqrt{10}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
Hence, the expressions that are equivalent to [tex]\(\sqrt{80}\)[/tex] are:
- [tex]\(80^{\frac{1}{2}}\)[/tex]
- [tex]\(4 \sqrt{5}\)[/tex]
