To determine which expression is equivalent to [tex]\(\frac{(a b^2)^3}{b^5}\)[/tex], let's simplify it step-by-step:
1. First, evaluate the numerator [tex]\((a b^2)^3\)[/tex]:
- Apply the power of a product rule: [tex]\((xy)^n = x^n y^n\)[/tex].
- Therefore, [tex]\((a b^2)^3 = a^3 (b^2)^3\)[/tex].
2. Simplify [tex]\((b^2)^3\)[/tex]:
- Use the power of a power rule: [tex]\((x^m)^n = x^{m \cdot n}\)[/tex].
- So, [tex]\((b^2)^3 = b^{2 \cdot 3} = b^6\)[/tex].
3. Substitute back into the expression:
- The numerator becomes [tex]\(a^3 b^6\)[/tex].
4. Now the expression is:
[tex]\[
\frac{a^3 b^6}{b^5}
\][/tex]
5. Simplify the fraction:
- Use the quotient of powers rule: [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex].
- So, [tex]\(\frac{b^6}{b^5} = b^{6-5} = b^1 = b\)[/tex].
6. Combine the results:
- The expression simplifies to [tex]\(a^3 b\)[/tex].
So, the equivalent expression is:
[tex]\[
a^3 b
\][/tex]
The correct answer is:
B. [tex]\(a^3 b\)[/tex]
