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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.

Which two expressions are equivalent to each other? (2^3)^4, 2^24 / 2^7, (2^2)^6, 1 / 2^12

Learn how to determine which two exponent expressions are equivalent by simplifying (2^3)^4, (2^24 / 2^7), (2^2)^6, and (1 / 2^12). This problem is designed for Grade 8 students and focuses on key exponent rules including power of a power and quotient of powers.

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