To find the expression equivalent to [tex]\( 13 \sqrt{22 b} - 10 \sqrt{22 b} \)[/tex], we can follow these step-by-step instructions:
1. Identify the Common Term: Both terms in the expression [tex]\( 13 \sqrt{22 b} - 10 \sqrt{22 b} \)[/tex] share a common factor, which is [tex]\( \sqrt{22 b} \)[/tex].
2. Factor Out the Common Term: We can simplify the expression by factoring out the common term, [tex]\( \sqrt{22 b} \)[/tex]. This means we are effectively grouping the terms together:
[tex]\[
13 \sqrt{22 b} - 10 \sqrt{22 b} = (13 - 10) \sqrt{22 b}
\][/tex]
3. Perform the Arithmetic Inside the Parentheses: Subtract 10 from 13 inside the parentheses:
[tex]\[
(13 - 10) \sqrt{22 b} = 3 \sqrt{22 b}
\][/tex]
4. Conclusion: After simplifying, we find that the equivalent expression is:
[tex]\[
3 \sqrt{22 b}
\][/tex]
In the provided options:
- Option A: [tex]\( 23 \sqrt{226} \)[/tex] does not match.
- Option B: [tex]\( 130 \sqrt{22 b} \)[/tex] does not match.
- Option C: [tex]\( 3 \sqrt{b^2} \)[/tex] does not match.
- Option D: [tex]\( 3 \sqrt{22 b} \)[/tex] is the correct match.
Thus, the correct answer is [tex]\( \boxed{D} \)[/tex].
