Let's solve the expression \((2^3)^3 \div (2^4)^4\).

1. Simplify \((2^3)^3\):
\[
(2^3)^3 = 2^{3 \times 3} = 2^9
\]

2. Simplify \((2^4)^4\):
\[
(2^4)^4 = 2^{4 \times 4} = 2^{16}
\]

3. Now, the expression becomes:
\[
\frac{2^9}{2^{16}}
\]

4. Apply the quotient rule for exponents, \( \frac{a^m}{a^n} = a^{m-n} \):
\[
2^{9-16} = 2^{-7}
\]

5. Finally, \(2^{-7}\) can also be written as:
\[
\frac{1}{2^7} = \frac{1}{128}
\]

So, \((2^3)^3 \div (2^4)^4 = \frac{1}{128}\).

Would you like more details on any step? Here are some related questions:

1. How do you simplify other exponent expressions like \((3^2)^4\)?
2. What is the rule for dividing expressions with the same base?
3. How can negative exponents be converted to fractions?
4. How do you handle exponents with different bases?
5. Can you explain the power rule of exponents in more detail?

**Tip:** When multiplying or dividing expressions with the same base, always use the exponent rules to simplify the calculation.

Learn how to simplify the expression (2³)³ ÷ (2⁴)⁴ using exponent rules, including the power of a power rule, the quotient rule, and negative exponents. This step-by-step solution is suitable for students in Grades 8-10 and provides a clear explanation of key concepts like exponentiation and simplifying expressions.

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