Which shows the following expression after the negative exponents have been eliminated? \( \frac{a^{3} b^{-2}}{a b^{-4}}, a \neq 0, b \neq 0 \) \( 0 \frac{a^{3} b^{-4}}{a b^{-2}} \) \( 0 \frac{a b^{4}}{a^{3} b^{2}} \) \( 0-\frac{a^{3} b^{4}}{a b^{2}} \) \( 0 \frac{a^{3} b^{4}}{a b^{2}} \)

When simplifying the expression \( \frac{a^{3} b^{-2}}{a b^{-4}} \), we can first address the negative exponents. Recall that \( b^{-n} = \frac{1}{b^{n}} \). Therefore, \( b^{-2} \) can be rewritten as \( \frac{1}{b^{2}} \) and \( b^{-4} \) as \( \frac{1}{b^{4}} \). This changes our expression to: \[ \frac{a^{3} \cdot \frac{1}{b^{2}}}{a \cdot \frac{1}{b^{4}}} = \frac{a^{3}}{b^{2}} \cdot \frac{b^{4}}{a} \] Now we can multiply it out: \[ \frac{a^{3} b^{4}}{a b^{2}} \] Now we can simplify: \[ = \frac{a^{3}}{a} \cdot \frac{b^{4}}{b^{2}} = a^{2} b^{2} \] So, the expression simplifies to \( a^{2} b^{2} \). None of the options listed in your question are correct since after simplification the negative exponents won't be there anymore. However, if you need a bit of fun facts: Did you know that exponents have been used since ancient civilizations, like the Babylonians? They would sometimes use a kind of shorthand in their calculations that resembles the way we think about exponents today! Now, when dealing with negative exponents, it’s common for students to forget that negative exponents can act like a reciprocal! A common mistake is to leave the negative exponent in the final answer, which creates unnecessary confusion—so always remember to flip those bad boys!