Solve ((10×3)^-4)^7: Complex Compound Exponent Expression

To solve the problem, we'll apply the exponent rule that states $\left(a^m\right)^n = a^{m \times n}$. Here’s how we proceed:

• Step 1: Recognize that the expression inside is $\left((10 \times 3)^{-4}\right)$, which is then raised to the 7th power.

• Step 2: Use the Power of a Power Rule: $\left(a^m\right)^n = a^{m \times n}$.

• Step 3: Applying this formula to our expression $\left((10 \times 3)^{-4}\right)^7$, results in $(10 \times 3)^{-4 \times 7}$.

• Step 4: Compute the multiplication in the exponent: $-4 \times 7 = -28$.

Therefore, $\left(\left(10\times3\right)^{-4}\right)^7 = (10 \times 3)^{-28}$.

Now, we need to compare our solution with the given choices:

• Choice 1: $(10 \times 3)^3$.

• Choice 2: $(10 \times 3)^{-11}$.

• Choice 3: $(10 \times 3)^{-28}$.

• Choice 4: $(10 \times 3)^{-\frac{7}{4}}$.

The correct choice is Choice 3: $(10 \times 3)^{-28}$, as this matches our simplified expression.