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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]\(\frac{a^4}{b}\)[/tex]

B. [tex]\(a^3 b\)[/tex]

C. [tex]\(\frac{a^3}{b}\)[/tex]

D. [tex]\(a^3\)[/tex]

Sagot :

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]