Questions: Which shows the following expression after the negative exponents have been eliminated? a^3 b^-2 / a b^-4, a ≠ 0, b ≠ 0 a^3 b^4 / a b^2 -a^3 b^4 / a b^2 a^3 b^-4 / a b^-2 a b^4 / a^3 b^2

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

a^3 b^-2 / a b^-4, a ≠ 0, b ≠ 0

a^3 b^4 / a b^2

-a^3 b^4 / a b^2

a^3 b^-4 / a b^-2

a b^4 / a^3 b^2

Transcript text: Which shows the following expression after the negative exponents have been eliminated? \[ \frac{a^{3} b^{-2}}{a b^{-4}}, a \neq 0, b \neq 0 \] $\frac{a^{3} b^{4}}{a b^{2}}$ $-\frac{a^{3} b^{4}}{a b^{2}}$ $\frac{a^{3} b^{-4}}{a b^{-2}}$ $\frac{a b^{4}}{a^{3} b^{2}}$

Solution

Solution Steps

To eliminate negative exponents, we can use the property that $ x^{-n} = \frac{1}{x^n} $. This means we can move terms with negative exponents from the numerator to the denominator or vice versa. Simplify the expression by applying this property and then simplify the resulting expression by canceling out common terms.

Step 1: Rewrite the Expression

We start with the expression: \[ \frac{a^{3} b^{-2}}{a b^{-4}} \] To eliminate the negative exponents, we can rewrite $ b^{-2} $ as $ \frac{1}{b^{2}} $ and $ b^{-4} $ as $ \frac{1}{b^{4}} $.

Step 2: Apply the Negative Exponent Rule

Using the property $ x^{-n} = \frac{1}{x^n} $, we can rewrite the expression as: \[ \frac{a^{3}}{a} \cdot \frac{b^{4}}{b^{2}} \]

Step 3: Simplify the Expression

Now, we simplify the expression: \[ \frac{a^{3}}{a} = a^{3-1} = a^{2} \] \[ \frac{b^{4}}{b^{2}} = b^{4-2} = b^{2} \] Thus, the simplified expression becomes: \[ a^{2} b^{2} \]