Let's analyze the given question and find out which two expressions are equivalent to each other.

The expressions to compare are:

1. $(2^3)^4$
2. $\frac{2^{24}}{2^7}$
3. $(2^2)^6$
4. $\frac{1}{2^{12}}$

First, let's simplify each expression:

1. $(2^3)^4$: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$

2. $\frac{2^{24}}{2^7}$: $\frac{2^{24}}{2^7} = 2^{24-7} = 2^{17}$

3. $(2^2)^6$: $(2^2)^6 = 2^{2 \cdot 6} = 2^{12}$

4. $\frac{1}{2^{12}}$: $\frac{1}{2^{12}} = 2^{-12}$

From these simplifications, we see that:

• $(2^3)^4 = 2^{12}$
• $\frac{2^{24}}{2^7} = 2^{17}$
• $(2^2)^6 = 2^{12}$
• $\frac{1}{2^{12}} = 2^{-12}$

Therefore, the two expressions that are equivalent to each other are $(2^3)^4$ and $(2^2)^6$.

Would you like any further details or explanations? Here are five related questions that expand on this information:

1. How do you simplify expressions with exponents involving multiplication and division?
2. What is the power of a power property in exponents?
3. How do you handle negative exponents?
4. Can you provide an example of simplifying a more complex exponent expression?
5. How would you convert a negative exponent to a positive exponent?

Tip: Remember, when dealing with exponents, the key properties include the product of powers, quotient of powers, and power of a power.