To simplify the expression [tex]\(( \sqrt{22} )( 5 \sqrt{2} )\)[/tex], let's follow the steps systematically:
1. Rewrite the expression:
[tex]\[
\sqrt{22} \cdot 5 \sqrt{2}
\][/tex]
2. Combine the square roots:
[tex]\[
\sqrt{22} \cdot 5 \sqrt{2} = 5 \cdot (\sqrt{22} \cdot \sqrt{2})
\][/tex]
3. Multiply the square roots:
[tex]\[
\sqrt{22} \cdot \sqrt{2} = \sqrt{22 \cdot 2} = \sqrt{44}
\][/tex]
4. Rewrite the expression with the simplified square root:
[tex]\[
5 \cdot \sqrt{44}
\][/tex]
5. Further simplify [tex]\(\sqrt{44}\)[/tex]:
[tex]\[
\sqrt{44} = \sqrt{4 \cdot 11} = \sqrt{4} \cdot \sqrt{11} = 2 \sqrt{11}
\][/tex]
6. Substitute [tex]\(\sqrt{44}\)[/tex] with its simplified form:
[tex]\[
5 \cdot 2 \sqrt{11}
\][/tex]
7. Multiply the constants:
[tex]\[
5 \cdot 2 \sqrt{11} = 10 \sqrt{11}
\][/tex]
Thus, the simplified form of [tex]\(( \sqrt{22} )( 5 \sqrt{2} )\)[/tex] is:
[tex]\[
\boxed{10 \sqrt{11}}
\][/tex]
So, the correct answer is:
[tex]\[
\text{A. } 10 \sqrt{11}
\][/tex]
