Certainly! Let's analyze each expression to determine if it is equivalent to [tex]\(\sqrt{40}\)[/tex].
1. First Expression: [tex]\(\sqrt{40}\)[/tex]
We start with the given expression:
[tex]\[
\sqrt{40}
\][/tex]
To simplify this, we need to look at the prime factors of 40:
[tex]\[
40 = 4 \times 10 = 2^2 \times 10
\][/tex]
Therefore,
[tex]\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}
\][/tex]
So, [tex]\(\sqrt{40} = 2\sqrt{10}\)[/tex]. This means [tex]\(\sqrt{40}\)[/tex] is equivalent to [tex]\(2\sqrt{10}\)[/tex].
2. Second Expression: [tex]\(40 \frac{1}{\frac{1}{2}}\)[/tex]
Simplify the fraction inside the expression:
[tex]\[
\frac{1}{\frac{1}{2}} = 2
\][/tex]
Therefore,
[tex]\[
40 \times 2 = 80
\][/tex]
Comparing this with our simplified [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
80 \neq 2\sqrt{10}
\][/tex]
So, this expression is not equivalent to [tex]\(\sqrt{40}\)[/tex].
3. Third Expression: [tex]\(2 \sqrt{10}\)[/tex]
We already know from our simplification that:
[tex]\[
\sqrt{40} = 2\sqrt{10}
\][/tex]
Therefore, [tex]\(2\sqrt{10}\)[/tex] is equivalent to [tex]\(\sqrt{40}\)[/tex].
4. Fourth Expression: [tex]\(5 \sqrt{8}\)[/tex]
Let's simplify [tex]\(5 \sqrt{8}\)[/tex]:
[tex]\[
8 = 4 \times 2 = 2^3
\][/tex]
Therefore,
[tex]\[
\sqrt{8} = \sqrt{2^3} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\][/tex]
So,
[tex]\[
5 \sqrt{8} = 5 \times 2 \sqrt{2} = 10 \sqrt{2}
\][/tex]
Comparing this with our simplified [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
10 \sqrt{2} \neq 2 \sqrt{10}
\][/tex]
Hence, this expression is not equivalent to [tex]\(\sqrt{40}\)[/tex].
5. Fifth Expression: [tex]\(4 \sqrt{10}\)[/tex]
Simplifying [tex]\(4 \sqrt{10}\)[/tex] as it is:
[tex]\[
4 \sqrt{10}
\][/tex]
Since we already have:
[tex]\[
2 \sqrt{10} \neq 4 \sqrt{10}
\][/tex]
So, this expression is not equivalent to [tex]\(\sqrt{40}\)[/tex].
6. Sixth Expression: [tex]\(160^{\frac{1}{2}}\)[/tex]
Simplify [tex]\(160^{\frac{1}{2}}\)[/tex]:
[tex]\[
\sqrt{160}
\][/tex]
We know:
[tex]\[
160 = 16 \times 10 = (4^2 \times 10)
\][/tex]
So,
[tex]\[
\sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10}
\][/tex]
Comparing this with our simplified [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
4 \sqrt{10} \neq 2 \sqrt{10}
\][/tex]
Therefore, this expression is not equivalent to [tex]\(\sqrt{40}\)[/tex].
Based on this detailed analysis, the expressions equivalent to [tex]\(\sqrt{40}\)[/tex] are:
[tex]\(2 \sqrt{10}\)[/tex].
