Solve Nested Powers: Evaluating ((4²)³)⁵

Insert the corresponding expression:

$\left(\left(4^2\right)^3\right)^5=$

To solve this problem, we need to simplify the expression $\left(\left(4^2\right)^3\right)^5$ using the rules of exponents.

Let's break down the problem step by step:

• Step 1: Apply the power of a power rule to $(4^2)^3$, which states that $(a^m)^n = a^{m \cdot n}$.
  Thus, $(4^2)^3 = 4^{2 \cdot 3} = 4^6$.
• Step 2: Now apply the power of a power rule again to the result we obtained in Step 1: $(4^6)^5$.
  Using the rule again, this becomes $(4^6)^5 = 4^{6 \cdot 5} = 4^{30}$.

Therefore, the simplified expression is $4^{30}$.

Let's verify against given choices:

• Choice 1: $4^{30}$ - This is correct.
• Choice 2: $4^{25}$ - Incorrect, results from missing a power multiplication.
• Choice 3: $4^{10}$ - Incorrect, results from only one power application.
• Choice 4: $4^{16}$ - Incorrect, possibly confusing the base calculation.

Thus, the correct answer is indeed $4^{30}$.