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The given expression is \( \left( 9^4 \right)^6 \), and we need to determine which of the provided options are equivalent to this expression.

Let's start simplifying \( \left( 9^4 \right)^6 \):

Using the power of a power rule for exponents, \( (a^m)^n = a^{m \cdot n} \), we have:

\[
\left( 9^4 \right)^6 = 9^{4 \cdot 6} = 9^{24}
\]

Now, let's evaluate the given options:

1. **\( 3^{24} \cdot 3^{24} \)**:
   Using the product of powers rule \( a^m \cdot a^n = a^{m+n} \), this becomes:
   \[
   3^{24} \cdot 3^{24} = 3^{24 + 24} = 3^{48}
   \]
   But \( 3^{48} \) is not equivalent to \( 9^{24} \).

2. **\( 9^{24} \)**:
   This is exactly the same as \( \left( 9^4 \right)^6 \), so it is **correct**.

3. **\( \frac{1}{9^{10}} \)**:
   This is not equivalent to \( 9^{24} \) because it represents a smaller exponent with a reciprocal, so it is **incorrect**.

4. **\( (9^6)^4 \)**:
   Using the same power of a power rule:
   \[
   (9^6)^4 = 9^{6 \cdot 4} = 9^{24}
   \]
   This is equivalent to \( 9^{24} \), so it is **correct**.

Thus, the correct equivalent expressions are:
- \( 9^{24} \)
- \( (9^6)^4 \)

Do you have any questions or would you like more details?

Here are five related questions to expand your understanding:
1. How would you simplify \( \left( 2^3 \right)^4 \)?
2. What is the product of \( 5^2 \cdot 5^3 \)?
3. How do you simplify \( \left( 4^5 \right)^2 \)?
4. Is \( \frac{1}{2^3} \) equivalent to \( 2^{-3} \)?
5. What is the general rule for multiplying powers of the same base?

**Tip:** Always check if powers can be simplified further using exponent rules like power of a power or product of powers.

Select all the expressions that are equivalent to (9^4)^6.

Learn how to simplify the expression (9^4)^6 using exponent rules. This problem demonstrates the application of the power of a power rule and checks for equivalency with expressions like 9^24 and (9^6)^4. Suitable for students in Grades 8-10.

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