Let's analyze each expression to determine which ones are equivalent to [tex]\(\sqrt{80}\)[/tex].
1. Expression: [tex]\(4 \sqrt{5}\)[/tex]
- To verify if this is equivalent to [tex]\(\sqrt{80}\)[/tex], we can check if they simplify to the same value.
[tex]\[
(4 \sqrt{5})^2 = (4^2 \cdot (\sqrt{5})^2) = 16 \cdot 5 = 80
\][/tex]
Since squaring [tex]\(4 \sqrt{5}\)[/tex] gives 80, it confirms that:
[tex]\[
4 \sqrt{5} = \sqrt{80}
\][/tex]
This expression is correct.
2. Expression: [tex]\(4 \sqrt{10}\)[/tex]
- To verify if this is equivalent to [tex]\(\sqrt{80}\)[/tex], we can again check the squared values.
[tex]\[
(4 \sqrt{10})^2 = (4^2 \cdot (\sqrt{10})^2) = 16 \cdot 10 = 160
\][/tex]
Squaring [tex]\(4 \sqrt{10}\)[/tex] gives 160, which is not equal to 80. Hence:
[tex]\[
4 \sqrt{10} \neq \sqrt{80}
\][/tex]
This expression is incorrect.
3. Expression: [tex]\(80^{\frac{1}{2}}\)[/tex]
- By definition, the square root of a number [tex]\(n\)[/tex] is equivalent to [tex]\(n^{\frac{1}{2}}\)[/tex].
[tex]\[
80^{\frac{1}{2}} = \sqrt{80}
\][/tex]
This expression is correct.
4. Expression: [tex]\(160^{\frac{1}{2}}\)[/tex]
- Calculate the square root of 160.
[tex]\[
160^{\frac{1}{2}} = \sqrt{160}
\][/tex]
Clearly, [tex]\(\sqrt{160}\)[/tex] is not equal to [tex]\(\sqrt{80}\)[/tex]. Hence:
[tex]\[
160^{\frac{1}{2}} \neq \sqrt{80}
\][/tex]
This expression is incorrect.
5. Expression: [tex]\(8 \sqrt{5}\)[/tex]
- Check the squared value of [tex]\(8\sqrt{5}\)[/tex].
[tex]\[
(8 \sqrt{5})^2 = (8^2 \cdot (\sqrt{5})^2) = 64 \cdot 5 = 320
\][/tex]
Squaring [tex]\(8 \sqrt{5}\)[/tex] gives 320, which is not equal to 80. Hence:
[tex]\[
8 \sqrt{5} \neq \sqrt{80}
\][/tex]
This expression is incorrect.
Conclusively, the expressions equivalent to [tex]\(\sqrt{80}\)[/tex] are:
[tex]\[
4 \sqrt{5} \text{ and } 80^{\frac{1}{2}}
\][/tex]
