jjinlov5
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Which shows the following expression after the negative exponents have been eliminated?

[tex]\[ \frac{a^3 b^{-2}}{a b^{-4}}, \quad a \neq 0, \quad b \neq 0 \][/tex]

A. [tex]\(\frac{a^3 b^{-4}}{a b^{-2}}\)[/tex]

B. [tex]\(\frac{a b^4}{a^3 b^2}\)[/tex]

C. [tex]\(-\frac{a^3 b^4}{a b^2}\)[/tex]

D. [tex]\(\frac{a^3 b^4}{a b^2}\)[/tex]



Answer :

GinnyAnswer
To simplify the given expression [tex]\(\frac{a^3 b^{-2}}{a b^{-4}}\)[/tex]:

1. Rewrite the negative exponents as positive exponents:

Recall that [tex]\( b^{-n} \)[/tex] can be written as [tex]\(\frac{1}{b^n}\)[/tex].

- [tex]\(b^{-2} = \frac{1}{b^2}\)[/tex]
- [tex]\(b^{-4} = \frac{1}{b^4}\)[/tex]

So, [tex]\(\frac{a^3 b^{-2}}{a b^{-4}}\)[/tex] becomes:
[tex]\[ \frac{a^3 \cdot \frac{1}{b^2}}{a \cdot \frac{1}{b^4}} = \frac{a^3 \cdot \frac{1}{b^2}}{a \cdot \frac{1}{b^4}} \][/tex]

2. Simplify the fractions inside the expression:

Multiply the numerator and the denominator by [tex]\(b^4\)[/tex] to eliminate the fraction in the denominator:
[tex]\[ \frac{a^3 \cdot \frac{1}{b^2}}{a \cdot \frac{1}{b^4}} = \frac{a^3 \cdot b^4 \cdot \frac{1}{b^2}}{a \cdot b^4 \cdot \frac{1}{b^4}} = \frac{a^3 \cdot b^4}{a \cdot b^2} \][/tex]

3. Combine and cancel common terms:

Simplify the expression:
[tex]\[ \frac{a^3 \cdot b^4}{a \cdot b^2} \][/tex]

- [tex]\(a^3\)[/tex] divided by [tex]\(a\)[/tex] is [tex]\(a^{3-1} = a^2\)[/tex]
- [tex]\(b^4\)[/tex] divided by [tex]\(b^2\)[/tex] is [tex]\(b^{4-2} = b^2\)[/tex]

So the simplified expression is:
[tex]\[ a^2 b^2 \][/tex]

Therefore, the correct final expression after eliminating the negative exponents and simplifying is:

[tex]\[ \frac{a^3 b^4}{a b^2} \][/tex]

The corresponding answer to the question is:

[tex]\[ \boxed{\frac{a^3 b^4}{a b^2}} \][/tex]

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