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.