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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.

1000^m ÷ 100^n can be written in the form 10^z. Express z in terms of m and n.

Learn how to express 1000^m ÷ 100^n in the form 10^z by using the laws of exponents. This problem focuses on simplifying powers of 10 and applying exponent rules. Suitable for students in Grades 9-11.

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