Select the correct answer.

Which expression is equivalent to the given expression? [tex]\frac{\left(a b^2\right)^3}{b^5}[/tex]

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]



Answer :

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]