To determine which expression is equivalent to [tex]\(\frac{(a b^2)^3}{b^5}\)[/tex], let's start by simplifying the given expression step-by-step.
1. Expand the numerator:
[tex]\[
(a b^2)^3
\][/tex]
When we raise a product to a power, every factor in the product is raised to the power separately. Therefore, we have:
[tex]\[
(a b^2)^3 = a^3 (b^2)^3
\][/tex]
2. Simplify the exponents:
We need to simplify [tex]\( (b^2)^3 \)[/tex]:
[tex]\[
(b^2)^3 = b^{2 \cdot 3} = b^6
\][/tex]
So, our numerator becomes:
[tex]\[
a^3 b^6
\][/tex]
3. Rewrite the expression with the simplified numerator:
The original expression now looks like this:
[tex]\[
\frac{a^3 b^6}{b^5}
\][/tex]
4. Simplify the fraction:
To simplify the fraction [tex]\(\frac{a^3 b^6}{b^5}\)[/tex], we can subtract the exponent of [tex]\(b\)[/tex] in the denominator from the exponent of [tex]\(b\)[/tex] in the numerator:
[tex]\[
\frac{b^6}{b^5} = b^{6 - 5} = b^1 = b
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
a^3 \cdot b
\][/tex]
So, the expression [tex]\(\frac{(a b^2)^3}{b^5}\)[/tex] simplifies to [tex]\(a^3 b\)[/tex].
Let's compare this with the given options:
A. [tex]\(a^3 b\)[/tex]
B. [tex]\(\frac{a^3}{b}\)[/tex]
C. [tex]\(\frac{a^4}{b}\)[/tex]
D. [tex]\(a^3\)[/tex]
We see that option A, [tex]\(a^3 b\)[/tex], matches our simplified expression. Therefore, the correct answer is:
A. [tex]\(a^3 b\)[/tex]
