Sure, let's solve the given expression step by step.
We need to simplify the expression
[tex]\[
\frac{(a \cdot b^2)^3}{b^5}
\][/tex]
### Step 1: Expand the numerator
First, we expand the expression in the numerator:
[tex]\[
(a \cdot b^2)^3
\][/tex]
To do this, we apply the power of a product rule, which states that [tex]\((xy)^n = x^n \cdot y^n\)[/tex]. Thus:
[tex]\[
(a \cdot b^2)^3 = a^3 \cdot (b^2)^3
\][/tex]
### Step 2: Simplify the exponents of [tex]\(b\)[/tex]
Next, we simplify [tex]\((b^2)^3\)[/tex] using the power of a power rule, which states that [tex]\((x^m)^n = x^{m \cdot n}\)[/tex]:
[tex]\[
(b^2)^3 = b^{2 \cdot 3} = b^6
\][/tex]
So the expression now becomes:
[tex]\[
a^3 \cdot b^6
\][/tex]
### Step 3: Substitute back into the original expression
We substitute [tex]\(a^3 \cdot b^6\)[/tex] back into the original fraction:
[tex]\[
\frac{a^3 \cdot b^6}{b^5}
\][/tex]
### Step 4: Simplify the fraction
To simplify the fraction, we use the quotient rule for exponents which states [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex]:
[tex]\[
\frac{a^3 \cdot b^6}{b^5} = a^3 \cdot b^{6-5} = a^3 \cdot b^1 = a^3 \cdot b
\][/tex]
So, the simplified form of the given expression [tex]\(\frac{(a b^2)^3}{b^5}\)[/tex] is:
[tex]\[
a^3 b
\][/tex]
### Step 5: Match the simplified expression with the options provided
Comparing with the provided options, we find:
- A. [tex]\(\frac{a^3}{b}\)[/tex]
- B. [tex]\(a^3 b\)[/tex]
- C. [tex]\(a^3\)[/tex]
- D. [tex]\(\frac{a^4}{b}\)[/tex]
The simplified expression [tex]\(a^3 b\)[/tex] matches option B.
Thus, the correct answer is:
[tex]\[
\boxed{B}
\][/tex]
