Evaluate the Double Power: (2⁷)⁵ Using Exponent Rules

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

• Step 1: Identify the given information.
• Step 2: Apply the appropriate exponent rule.
• Step 3: Perform the necessary calculations.

Let's work through each step:

Step 1: The given expression is $\left(2^7\right)^5$. Here, the base is $2$, and we have two exponents: $7$ in the inner expression and $5$ outside.

Step 2: We'll use the power of a power rule for exponents, which states $(a^m)^n = a^{m \cdot n}$. This means we will multiply the exponents $7$ and $5$.

Step 3: Calculating, we multiply the exponents:
$7 \times 5 = 35$

Therefore, the expression $\left(2^7\right)^5$ simplifies to $2^{35}$.

Now, let's verify with the given answer choices:

• Choice 1: $2^{12}$ - Incorrect, as the exponents were not multiplied properly.
• Choice 2: $2^2$ - Incorrect, as it significantly underestimates the combined exponent value.
• Choice 3: $2^{35}$ - Correct, matches the calculated exponent.
• Choice 4: $2^{\frac{5}{7}}$ - Incorrect, involves incorrect fraction of exponents.

Thus, the correct choice is Choice 3: $2^{35}$.

I am confident in the correctness of this solution as it directly applies well-established exponent rules.