Solve Nested Exponents: ((10×7)^8)^6 Step-by-Step Solution

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

• Step 1: Identify the given expression.
• Step 2: Apply the "power of a power" rule for exponents.
• Step 3: Perform the necessary calculations to simplify the expression.

Now, let's work through each step:

Step 1: Identify the expression $\left(\left(10 \times 7\right)^8\right)^6$.
Step 2: Using the "power of a power" theorem, which states $(a^m)^n = a^{m \cdot n}$, we apply this to the expression.
Step 3: Inside our expression, $a = 10 \times 7$, $m = 8$, and $n = 6$. Thus, $\left(\left(10 \times 7\right)^8\right)^6$ becomes $\left(10 \times 7\right)^{8 \times 6}$.

Step 4: Calculate the new exponent: $8 \times 6 = 48$. Thus, the expression simplifies to $\left(10 \times 7\right)^{48}$.

Therefore, the solution to the problem is $\left(10 \times 7\right)^{48}$.

Now, let's consider the provided answer choices:

• Choice 1: $\left(10\times7\right)^2$ - Incorrect because this does not align with our calculation.
• Choice 2: $\left(10\times7\right)^{14}$ - Incorrect because the exponent is not calculated as $8 \times 6$.
• Choice 3: $\left(10\times7\right)^{\frac{2}{4}}$ - Incorrect because this exponent is not what results from $8 \times 6$.
• Choice 4: $\left(10\times7\right)^{48}$ - Correct, as it matches our simplified result.

We conclude that the correct solution is option 4.