1. **State the problem:** We want to express $\frac{100^m}{100^n}$ in the form $10^z$ and find $z$ in terms of $m$ and $n$. 2. **Recall the properties of exponents:** - $\frac{a^m}{a^n} = a^{m-n}$ for any base $a$. - $100 = 10^2$. 3. **Rewrite the expression using base 10:** $$\frac{100^m}{100^n} = \frac{(10^2)^m}{(10^2)^n}$$ 4. **Apply the power of a power rule:** $$(10^2)^m = 10^{2m} \quad \text{and} \quad (10^2)^n = 10^{2n}$$ 5. **Substitute back:** $$\frac{10^{2m}}{10^{2n}} = 10^{2m - 2n}$$ 6. **Simplify the exponent:** $$10^{2(m-n)}$$ 7. **Conclusion:** The expression $\frac{100^m}{100^n}$ can be written as $10^z$ where $$z = 2(m - n)$$