Mastering Negative Exponents: Simplifying (ax/b)⁻ʐ

To solve this problem, we first recognize that we have the expression $(\frac{ax}{b})^{-z}$. Our goal is to rewrite this with positive exponents.

Step 1: Apply the negative exponent rule. For any non-zero base $y$, $y^{-n} = \frac{1}{y^n}$. Hence, $(\frac{ax}{b})^{-z} = \frac{1}{(\frac{ax}{b})^z}$.

Step 2: Rewrite the expression using the property of exponents for fractions. For $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$, we get $\frac{1}{(\frac{ax}{b})^z} = \frac{1}{\frac{(ax)^z}{b^z}} = \frac{b^z}{(ax)^z}$.

Step 3: Express the power on $ax$. The expression $(ax)^z$ becomes $a^z \cdot x^z$.

Step 4: Substitute back into the expression. We have $\frac{b^z}{a^z \cdot x^z}$ which is $b^z \cdot a^{-z} \cdot x^{-z}$.

Therefore, the expression $(\frac{ax}{b})^{-z}$ simplifies to $b^z a^{-z} x^{-z}$.

Upon comparing this result with the provided answer choices, we see that it matches the option labeled as choice 2: $b^z a^{-z} x^{-z}$.

Therefore, the solution to the problem is $b^z a^{-z} x^{-z}$.