To determine which expressions are equivalent to [tex]\(\sqrt{40}\)[/tex], we will evaluate each given expression step-by-step.
1. Expression: [tex]\(160^{\frac{1}{3}}\)[/tex]
Let's simplify [tex]\(160^{\frac{1}{3}}\)[/tex]:
- This expression represents the cube root of 160.
- Clearly, the cube root of 160 does not simplify to match [tex]\(\sqrt{40}\)[/tex], therefore [tex]\(160^{\frac{1}{3}}\)[/tex] is not equivalent to [tex]\(\sqrt{40}\)[/tex].
2. Expression: [tex]\(40^{\frac{1}{3}}\)[/tex]
Let's simplify [tex]\(40^{\frac{1}{3}}\)[/tex]:
- This expression represents the cube root of 40.
- Again, the cube root of 40 does not simplify to match [tex]\(\sqrt{40}\)[/tex], therefore [tex]\(40^{\frac{1}{3}}\)[/tex] is not equivalent to [tex]\(\sqrt{40}\)[/tex].
3. Expression: [tex]\(2 \sqrt{10}\)[/tex]
Let's simplify [tex]\(2 \sqrt{10}\)[/tex]:
- [tex]\(\sqrt{40}\)[/tex] can be written as [tex]\(\sqrt{4 \times 10}\)[/tex].
- [tex]\(\sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10}\)[/tex].
- Hence, [tex]\(2 \sqrt{10}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex].
4. Expression: [tex]\(5 \sqrt{8}\)[/tex]
Let's simplify [tex]\(5 \sqrt{8}\)[/tex]:
- [tex]\(\sqrt{8}\)[/tex] can be written as [tex]\(\sqrt{4 \times 2} = 2 \sqrt{2}\)[/tex].
- So, [tex]\(5 \sqrt{8} = 5 \times 2 \sqrt{2} = 10 \sqrt{2}\)[/tex].
- Clearly, [tex]\(10 \sqrt{2}\)[/tex] does not simplify to match [tex]\(\sqrt{40}\)[/tex], therefore [tex]\(5 \sqrt{8}\)[/tex] is not equivalent to [tex]\(\sqrt{40}\)[/tex].
In summary, only the expression [tex]\(2 \sqrt{10}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex].
