Solve ((a×b)³)⁷: Simplifying Nested Compound Exponents

Let's solve the problem by applying the steps outlined in the analysis.

• Step 1: Identify the expression we need to simplify: $\left(\left(a \times b\right)^3\right)^7$.

• Step 2: Apply the power of a power rule ($\left(x^m\right)^n = x^{m \times n}$) to the entire expression.

Apply the rule:
$\left(\left(a \times b\right)^3\right)^7 = \left(a \times b\right)^{3 \times 7}$ This simplifies to: $\left(a \times b\right)^{21}$

The expression simplifies to $\left(a \times b\right)^{21}$.

Now, let's consider the choices:

• Choice 1: $\left(a \times b\right)^{21}$ is correct, as it matches the result of our simplification.

• Choice 2: $\left(a \times b\right)^{3-7}$ is incorrect, as it incorrectly subtracts the exponents instead of multiplying them.

• Choice 3: $\left(a \times b\right)^{7+3}$ is incorrect, as it incorrectly adds the exponents instead of multiplying them.

• Choice 4: $\left(a \times b\right)^{\frac{7}{3}}$ is incorrect, as it applies division instead of multiplication to the exponents.

Therefore, the correct choice is Choice 1: $\left(a \times b\right)^{21}$.