To express $1000^m \div 100^n$ in the form $10^z$, let's follow these steps:

Step 1: Rewrite 1000 and 100 as powers of 10.

• $1000 = 10^3$ (since $1000 = 10^3$).
• $100 = 10^2$ (since $100 = 10^2$).

Step 2: Substitute these into the expression.

We now have: $1000^m \div 100^n = (10^3)^m \div (10^2)^n.$

Step 3: Apply exponent rules.

• When raising a power to a power, multiply the exponents: $(10^3)^m = 10^{3m}$ and $(10^2)^n = 10^{2n}$.
• When dividing powers of the same base, subtract the exponents: $\frac{10^{3m}}{10^{2n}} = 10^{3m - 2n}.$

Step 4: Express in the form $10^z$.

Thus, we can express $z$ as: $z = 3m - 2n.$

So, the value of $z$ in terms of $m$ and $n$ is $z = 3m - 2n$.

Would you like any further details on this process?

Here are five related questions:

1. How would this change if the bases were different, such as $5$ instead of $10$?
2. What happens if you reverse the operation to $100^n \div 1000^m$?
3. How can the properties of logarithms help solve such problems?
4. What if the powers are fractional, such as $1000^{m/2}$?
5. Can you extend this to an expression involving higher powers of 10, such as $10000^p$?

Tip: Always remember that when dealing with powers of the same base, multiplication adds the exponents, and division subtracts them.