Answer :
Let's consider the given expression, [tex]\(\sqrt{40}\)[/tex], and compare it to the other expressions to determine if they are equivalent.
1. Expression: [tex]\(5 \sqrt{8}\)[/tex]
- To determine if [tex]\(5 \sqrt{8}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex], we simplify [tex]\(5 \sqrt{8}\)[/tex]:
[tex]\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2} \][/tex]
Therefore,
[tex]\[ 5 \sqrt{8} = 5 \times 2 \sqrt{2} = 10 \sqrt{2} \][/tex]
Comparing [tex]\(10 \sqrt{2}\)[/tex] to [tex]\(\sqrt{40}\)[/tex], we see they are not equal because [tex]\(\sqrt{40}\)[/tex] simplifies to [tex]\(2 \sqrt{10}\)[/tex]. Hence, [tex]\(5 \sqrt{8} \neq \sqrt{40}\)[/tex]. So, [tex]\(5 \sqrt{8}\)[/tex] is not equivalent.
2. Expression: [tex]\(40^{\frac{1}{2}}\)[/tex]
- We can express the square root of 40 equivalently as:
[tex]\[ 40^{\frac{1}{2}} = \sqrt{40} \][/tex]
Hence, [tex]\(40^{\frac{1}{2}} = \sqrt{40}\)[/tex]. So, [tex]\(40^{\frac{1}{2}}\)[/tex] is equivalent.
3. Expression: [tex]\(4 \sqrt{10}\)[/tex]
- To determine if [tex]\(4 \sqrt{10}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex], consider the simplification of [tex]\(\sqrt{40}\)[/tex]:
[tex]\[ \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10} \][/tex]
Comparing [tex]\(4 \sqrt{10}\)[/tex] to [tex]\(2 \sqrt{10}\)[/tex], we observe they are not equal. Hence, [tex]\(4 \sqrt{10} \neq \sqrt{40}\)[/tex]. So, [tex]\(4 \sqrt{10}\)[/tex] is not equivalent.
4. Expression: [tex]\(160^{\frac{1}{2}}\)[/tex]
- Considering [tex]\(160^{\frac{1}{2}}\)[/tex] as the square root of 160:
[tex]\[ 160^{\frac{1}{2}} = \sqrt{160} \][/tex]
Simplify [tex]\(\sqrt{160}\)[/tex]:
[tex]\[ \sqrt{160} = \sqrt{16 \cdot 10} = \sqrt{16} \cdot \sqrt{10} = 4 \sqrt{10} \][/tex]
Comparing [tex]\(4 \sqrt{10}\)[/tex] to [tex]\(\sqrt{40}\)[/tex], which is [tex]\(2 \sqrt{10}\)[/tex], we see they are not equal. Hence, [tex]\(160^{\frac{1}{2}} \neq \sqrt{40}\)[/tex]. So, [tex]\(160^{\frac{1}{2}}\)[/tex] is not equivalent.
5. Expression: [tex]\(2 \sqrt{10}\)[/tex]
- Considering [tex]\(2 \sqrt{10}\)[/tex] and simplifying [tex]\(\sqrt{40}\)[/tex]:
[tex]\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \][/tex]
Since [tex]\(2 \sqrt{10}\)[/tex] equals [tex]\(\sqrt{40}\)[/tex], they are indeed equivalent.
Summarizing the results:
- [tex]\(40^{\frac{1}{2}}\)[/tex] and [tex]\(2 \sqrt{10}\)[/tex] are equivalent to [tex]\(\sqrt{40}\)[/tex].
- [tex]\(5 \sqrt{8}\)[/tex], [tex]\(4 \sqrt{10}\)[/tex], and [tex]\(160^{\frac{1}{2}}\)[/tex] are not equivalent to [tex]\(\sqrt{40}\)[/tex].
The correct expressions equivalent to [tex]\(\sqrt{40}\)[/tex] are:
[tex]\[ 40^{\frac{1}{2}}, \quad 2 \sqrt{10} \][/tex]
1. Expression: [tex]\(5 \sqrt{8}\)[/tex]
- To determine if [tex]\(5 \sqrt{8}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex], we simplify [tex]\(5 \sqrt{8}\)[/tex]:
[tex]\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2} \][/tex]
Therefore,
[tex]\[ 5 \sqrt{8} = 5 \times 2 \sqrt{2} = 10 \sqrt{2} \][/tex]
Comparing [tex]\(10 \sqrt{2}\)[/tex] to [tex]\(\sqrt{40}\)[/tex], we see they are not equal because [tex]\(\sqrt{40}\)[/tex] simplifies to [tex]\(2 \sqrt{10}\)[/tex]. Hence, [tex]\(5 \sqrt{8} \neq \sqrt{40}\)[/tex]. So, [tex]\(5 \sqrt{8}\)[/tex] is not equivalent.
2. Expression: [tex]\(40^{\frac{1}{2}}\)[/tex]
- We can express the square root of 40 equivalently as:
[tex]\[ 40^{\frac{1}{2}} = \sqrt{40} \][/tex]
Hence, [tex]\(40^{\frac{1}{2}} = \sqrt{40}\)[/tex]. So, [tex]\(40^{\frac{1}{2}}\)[/tex] is equivalent.
3. Expression: [tex]\(4 \sqrt{10}\)[/tex]
- To determine if [tex]\(4 \sqrt{10}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex], consider the simplification of [tex]\(\sqrt{40}\)[/tex]:
[tex]\[ \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10} \][/tex]
Comparing [tex]\(4 \sqrt{10}\)[/tex] to [tex]\(2 \sqrt{10}\)[/tex], we observe they are not equal. Hence, [tex]\(4 \sqrt{10} \neq \sqrt{40}\)[/tex]. So, [tex]\(4 \sqrt{10}\)[/tex] is not equivalent.
4. Expression: [tex]\(160^{\frac{1}{2}}\)[/tex]
- Considering [tex]\(160^{\frac{1}{2}}\)[/tex] as the square root of 160:
[tex]\[ 160^{\frac{1}{2}} = \sqrt{160} \][/tex]
Simplify [tex]\(\sqrt{160}\)[/tex]:
[tex]\[ \sqrt{160} = \sqrt{16 \cdot 10} = \sqrt{16} \cdot \sqrt{10} = 4 \sqrt{10} \][/tex]
Comparing [tex]\(4 \sqrt{10}\)[/tex] to [tex]\(\sqrt{40}\)[/tex], which is [tex]\(2 \sqrt{10}\)[/tex], we see they are not equal. Hence, [tex]\(160^{\frac{1}{2}} \neq \sqrt{40}\)[/tex]. So, [tex]\(160^{\frac{1}{2}}\)[/tex] is not equivalent.
5. Expression: [tex]\(2 \sqrt{10}\)[/tex]
- Considering [tex]\(2 \sqrt{10}\)[/tex] and simplifying [tex]\(\sqrt{40}\)[/tex]:
[tex]\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \][/tex]
Since [tex]\(2 \sqrt{10}\)[/tex] equals [tex]\(\sqrt{40}\)[/tex], they are indeed equivalent.
Summarizing the results:
- [tex]\(40^{\frac{1}{2}}\)[/tex] and [tex]\(2 \sqrt{10}\)[/tex] are equivalent to [tex]\(\sqrt{40}\)[/tex].
- [tex]\(5 \sqrt{8}\)[/tex], [tex]\(4 \sqrt{10}\)[/tex], and [tex]\(160^{\frac{1}{2}}\)[/tex] are not equivalent to [tex]\(\sqrt{40}\)[/tex].
The correct expressions equivalent to [tex]\(\sqrt{40}\)[/tex] are:
[tex]\[ 40^{\frac{1}{2}}, \quad 2 \sqrt{10} \][/tex]
