To simplify the given expression, [tex]\((\sqrt{22})(5 \sqrt{2})\)[/tex], we will proceed through several steps:
1. Combine the constants and the square root parts separately:
[tex]\[
\sqrt{22} \times 5 \sqrt{2} = 5 \times (\sqrt{22} \times \sqrt{2})
\][/tex]
2. Use the property of square roots:
The property [tex]\(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)[/tex] allows us to combine the square roots:
[tex]\[
\sqrt{22} \times \sqrt{2} = \sqrt{22 \times 2}
\][/tex]
3. Perform the multiplication inside the square root:
[tex]\[
22 \times 2 = 44
\][/tex]
So, the expression becomes:
[tex]\[
5 \times \sqrt{44}
\][/tex]
4. Simplify [tex]\(\sqrt{44}\)[/tex] further:
Notice that [tex]\(44\)[/tex] can be factored into [tex]\(4 \times 11\)[/tex], thus:
[tex]\[
\sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2 \sqrt{11}
\][/tex]
5. Substitute back and multiply the values:
[tex]\[
5 \times \sqrt{44} = 5 \times (2 \sqrt{11}) = 10 \sqrt{11}
\][/tex]
Thus, the simplified form of the given expression [tex]\((\sqrt{22})(5 \sqrt{2})\)[/tex] is:
[tex]\[
10 \sqrt{11}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{10 \sqrt{11}}
\][/tex]
The choice corresponds to:
A. [tex]\(10 \sqrt{11}\)[/tex]
