Solve (10×2)^7)^3: Nested Exponents Challenge

To solve this problem, we'll simplify the expression $(\left(10 \times 2\right)^7)^3$ using the rules of exponents:

• Step 1: Identify the structure of the expression
• Step 2: Apply the power of a power rule
• Step 3: Verify the correctness of our answer with the provided choices

Now, let's work through each step:

Step 1: The expression $(\left(10 \times 2\right)^7)^3$ involves two operations: the multiplication inside the parentheses and the power raised to another power outside.

Step 2: We use the power of a power rule $(a^m)^n = a^{m \times n}$. Applying this to the base $\left(10 \times 2\right)$ and the exponents 7 and 3, we have:

$(\left(10 \times 2\right)^7)^3 = \left(10 \times 2\right)^{7 \times 3}$

This simplifies further to:

$\left(10 \times 2\right)^{21}$

Step 3: Now, let's verify with the given choices:
- Choice 1: $\left(10 \times 2\right)^{7+3}$, incorrect because it applies addition instead of multiplication of exponents.
- Choice 2: $\left(10 \times 2\right)^{7 \times 3}$, correct, as it correctly follows the power of a power rule.
- Choice 3: $\left(10 \times 2\right)^{7-3}$, incorrect because it subtracts exponents.
- Choice 4: $\left(10 \times 2\right)^{\frac{3}{7}}$, incorrect because it divides the exponents.

Therefore, the correct choice is Choice 2: $\left(10 \times 2\right)^{7 \times 3}$.

Hence, the simplified expression is $\left(10 \times 2\right)^{21}$.