Explore IDNLearn.com's extensive Q&A database and find the answers you're looking for. Our community provides timely and precise responses to help you understand and solve any issue you face.
Sagot :
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]
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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.