To determine which expressions are equivalent to [tex]\(\sqrt{40}\)[/tex], we will simplify [tex]\(\sqrt{40}\)[/tex] and compare it with the given expressions. Here is the step-by-step solution:
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10}
\][/tex]
Thus, [tex]\(\sqrt{40}\)[/tex] simplifies to [tex]\(2 \sqrt{10}\)[/tex].
2. Compare with the given expressions one by one:
- Expression 1: [tex]\(4 \sqrt{10}\)[/tex]
[tex]\[
4 \sqrt{10}
\][/tex]
This is not equivalent to [tex]\(2 \sqrt{10}\)[/tex].
- Expression 2: [tex]\(160^{\frac{1}{2}}\)[/tex]
[tex]\[
160^{\frac{1}{2}} = \sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4 \sqrt{10}
\][/tex]
This is not equivalent to [tex]\(2 \sqrt{10}\)[/tex].
- Expression 3: [tex]\(40^{\frac{1}{2}}\)[/tex]
[tex]\[
40^{\frac{1}{2}} = \sqrt{40}
\][/tex]
This is equivalent to [tex]\(2 \sqrt{10}\)[/tex].
- Expression 4: [tex]\(2 \sqrt{10}\)[/tex]
[tex]\[
2 \sqrt{10}
\][/tex]
This is equivalent to [tex]\(2 \sqrt{10}\)[/tex].
- Expression 5: [tex]\(5 \sqrt{8}\)[/tex]
[tex]\[
5 \sqrt{8} = 5 \sqrt{4 \times 2} = 5 \times \sqrt{4} \times \sqrt{2} = 5 \times 2 \sqrt{2} = 10 \sqrt{2}
\][/tex]
This is not equivalent to [tex]\(2 \sqrt{10}\)[/tex].
Therefore, the expressions that are equivalent to [tex]\(\sqrt{40}\)[/tex] are:
[tex]\[
40^{\frac{1}{2}} \quad \text{and} \quad 2 \sqrt{10}
\][/tex]
In summary, the correct expressions are:
[tex]\[ \boxed{3, 4} \][/tex]
