The expression you're working with is:

\[
\frac{4^3}{4^7}
\]

To simplify this, we can apply the rule of exponents: 

\[
\frac{a^m}{a^n} = a^{m-n}
\]

So, for this expression:

\[
\frac{4^3}{4^7} = 4^{3-7} = 4^{-4}
\]

A negative exponent means the reciprocal, so:

\[
4^{-4} = \frac{1}{4^4}
\]

Now, calculate \(4^4\):

\[
4^4 = 4 \times 4 \times 4 \times 4 = 256
\]

Thus, the simplified result is:

\[
\frac{1}{256}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do you simplify \(\frac{a^m}{a^n}\) when \(a\) is any constant?
2. What is the value of \(2^{-5}\)?
3. How would you express \(x^{-3}\) in a positive exponent form?
4. How do negative exponents apply to fractional bases like \(\frac{1}{2}\)?
5. What is the result of \(3^0\) and why?

**Tip:** Always remember that a negative exponent flips the base into its reciprocal.

Learn how to simplify the expression 4^3 divided by 4^7 using the rules of exponents. This detailed explanation breaks down the exponentiation process, showing how to calculate 4^(-4) and find the final result of 1/256. Suitable for students in Grades 6-8.

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