Simplify ((ax)^7)^2: Nested Exponent Expression Solution

To solve this problem, we'll follow these steps:

• Identify the given information: We have the expression $(\left(a \times x\right)^7)^2$.
• Apply the appropriate formula: Use the "power of a power" rule for exponents.
• Perform the necessary calculations: Simplify the expression using the rule.

Now, let's work through each step:
Step 1: The expression given is $(\left(a \times x\right)^7)^2$.
Step 2: According to the power of a power rule, $(b^m)^n = b^{m \times n}$. Hence, $(\left(a \times x\right)^7)^2 = (\left(a \times x\right)^{7 \times 2})$.
Step 3: Perform the multiplication in the exponent, which results in $(\left(a \times x\right)^{14})$.

Therefore, the expression $(\left(a \times x\right)^7)^2$ simplifies to $(a \times x)^{14}$.

Checking against the answer choices:

• Choice 1: $\left(a\times x\right)^{7-2}$ simplifies to $\left(a\times x\right)^5$. Incorrect.
• Choice 2: $\left(a\times x\right)^{\frac{2}{7}}$. Incorrect application of exponent rules.
• Choice 3: $\left(a\times x\right)^{7+2} = \left(a\times x\right)^9$. Incorrect.
• Choice 4: $\left(a\times x\right)^{7\times2} = \left(a\times x\right)^{14}$. Correct.

Given all these considerations, the correct choice is Choice 4: $\left(a\times x\right)^{14}$, which corresponds to the correct application of the "power of a power" rule.