\( \frac { a ^ { 3 } b ^ { - 2 } } { a b ^ { - 4 } } , a \neq 0 , b \neq 0 \)

To simplify the expression \( \frac { a ^ { 3 } b ^ { - 2 } } { a b ^ { - 4 } } \), we can start by using the properties of exponents. When dividing like bases, you subtract the exponents. The expression can be rewritten as: \[ = a^{3 - 1} b^{-2 - (-4)} \] \[ = a^{2} b^{-2 + 4} \] \[ = a^{2} b^{2} \] Thus, the simplified form is \( a^{2} b^{2} \), or \( \frac{a^2}{b^2} \) if we decide to express it with positive exponents. Now, when it comes to working with fractions in algebra, it's important to keep track of negatives and remember to apply the rules of exponents carefully. A common mistake is misapplying the subtraction of the exponents, which can lead to incorrect simplifications! Always double-check those exponents!