Solve for X in ((4×6)⁴)ˣ: Nested Exponent Expression

To solve this problem, we will simplify the expression $\left(\left(4 \times 6\right)^4\right)^x$ by following these steps:

• Step 1: Identify the base and exponents. The base is $4 \times 6$ and the inner exponent is $4$, which is then raised to $x$.
• Step 2: Apply the power of a power rule $(a^m)^n = a^{m \cdot n}$. We apply this rule to $\left(\left(4 \times 6\right)^4\right)^x$.
• Step 3: Perform the multiplication of exponents. $(4 \times 6)^{4 \cdot x} = (4 \times 6)^{4x}$.

Now, let's work through each of these steps:

Step 1: We are given the compound expression $\left(\left(4 \times 6\right)^4\right)^x$. The base here is $4 \times 6$, the first exponent is 4, and the second exponent is $x$.

Step 2: Using the formula for a power of a power, we have $(a^m)^n = a^{m \cdot n}$. Substitute the values: $(4 \times 6)^4$ is being raised to $x$.

Step 3: Simplify the expression: We then multiply the exponents $4 \times x$ to get $(4 \times 6)^{4x}$.

Thus, the expression simplifies to $(4 \times 6)^{4x}$.

Reviewing the given choices:

• Choice 1: $\left(4 \times 6\right)^{4+x}$ - Incorrect, as it adds the exponents.
• Choice 2: $\left(4 \times 6\right)^{4-x}$ - Incorrect, as it subtracts the exponents.
• Choice 3: $\left(4 \times 6\right)^{4x}$ - Correct, as it correctly applies the power of a power rule.
• Choice 4: $\left(4 \times 6\right)^{\frac{x}{4}}$ - Incorrect, as it incorrectly applies division to the exponents.

Therefore, the correct choice is choice 3: $(4 \times 6)^{4x}$.