To find which expressions are equivalent to [tex]\(\sqrt{80}\)[/tex], we need to simplify or evaluate each of the given expressions. Let's do this step-by-step:
1. Expression: [tex]\(80^{\frac{1}{2}}\)[/tex]
This is another way of writing the square root of 80. So:
[tex]\[
80^{\frac{1}{2}} = \sqrt{80}
\][/tex]
Thus, [tex]\(80^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(\sqrt{80}\)[/tex].
2. Expression: [tex]\(4 \sqrt{5}\)[/tex]
Let's see if this expression equals [tex]\(\sqrt{80}\)[/tex]:
[tex]\[
4 \sqrt{5} = 4 \times \sqrt{5}
\][/tex]
Squaring both sides to compare with 80:
[tex]\[
(4 \sqrt{5})^2 = 16 \times 5 = 80
\][/tex]
Taking the square root of 80 matches the original expression. Therefore, [tex]\(4 \sqrt{5}\)[/tex] is equivalent to [tex]\(\sqrt{80}\)[/tex].
3. Expression: [tex]\(4 \sqrt{10}\)[/tex]
Let's simplify and see if this equals [tex]\(\sqrt{80}\)[/tex]:
[tex]\[
4 \sqrt{10} = 4 \times \sqrt{10}
\][/tex]
Squaring both sides:
[tex]\[
(4 \sqrt{10})^2 = 16 \times 10 = 160
\][/tex]
Since [tex]\( \sqrt{160} \neq \sqrt{80} \)[/tex], the expression [tex]\(4 \sqrt{10}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
4. Expression: [tex]\(160^{\frac{1}{2}}\)[/tex]
This is the square root of 160:
[tex]\[
160^{\frac{1}{2}} = \sqrt{160}
\][/tex]
Since [tex]\( \sqrt{160} \neq \sqrt{80} \)[/tex], the expression [tex]\(160^{\frac{1}{2}}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
5. Expression: [tex]\(8 \sqrt{5}\)[/tex]
Simplify this expression:
[tex]\[
8 \sqrt{5} = 8 \times \sqrt{5}
\][/tex]
Squaring both sides:
[tex]\[
(8 \sqrt{5})^2 = 64 \times 5 = 320
\][/tex]
Since [tex]\( \sqrt{320} \neq \sqrt{80} \)[/tex], the expression [tex]\(8 \sqrt{5}\)[/tex] is not equivalent to [tex]\(\sqrt{80}\)[/tex].
Given the above simplifications and comparisons, 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]\( 80^{\frac{1}{2}} \)[/tex]
- [tex]\( 4 \sqrt{5} \)[/tex]
