Solve Nested Exponents: ((7a)^4)^3 Using Laws of Powers

To solve this problem, we'll simplify the expression $\left(\left(7\times a\right)^4\right)^3$ using the rules of exponents, specifically the power of a power rule.

Let's follow these steps:

• Step 1: Identify the structure of the expression
• Step 2: Apply the power of a power rule
• Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The expression given is $\left(\left(7 \times a\right)^4\right)^3$. This is an example of a power raised to another power.

Step 2: According to the power of a power rule, $(b^m)^n = b^{m \times n}$, we multiply the exponents. Here, the base is $7 \times a$, the first exponent (m) is 4, and the second exponent (n) is 3.

Step 3: Multiply the exponents:

$m \times n = 4 \times 3 = 12$

Thus, the expression simplifies to $(7 \times a)^{12}$.

Therefore, the solution to the problem is $\left(7 \times a\right)^{12}$.

Finally, let's verify our solution against the provided choices:

• Choice 1: $\left(7\times a\right)^1$ - This is incorrect because the exponents aren't simplified correctly.
• Choice 2: $\left(7\times a\right)^7$ - This is incorrect for the same reason.
• Choice 3: $\left(7\times a\right)^{12}$ - This matches our simplified solution.
• Choice 4: $\left(7\times a\right)^{\frac{3}{4}}$ - This doesn't match because the power of a power rule doesn't lead to fractional exponents in this problem.

Hence, Choice 3 is the correct choice and the answer is $\left(7 \times a\right)^{12}$.