Solve ((3×5)^-3)^-6: Nested Negative Exponents Challenge

Insert the corresponding expression:

$\left(\left(3\times5\right)^{-3}\right)^{-6}=$

To simplify the expression $\left(\left(3\times5\right)^{-3}\right)^{-6}$, we apply exponent rules, specifically the power of a power rule.

Here's a step-by-step solution:

• Step 1: Identify the structure of the expression: we have an outer exponent $-6$ and an inner exponent $-3$ applied to the base $(3 \times 5)$.

• Step 2: Apply the power of a power rule, which states $(a^m)^n = a^{m \cdot n}$. This combines the exponents being multiplied.

• Step 3: Calculate the multiplication of the exponents: $-3 \times -6$. This yields $18$ since multiplying two negative numbers results in a positive number.

• Step 4: Substitute back into the expression: $\left(3 \times 5\right)^{18}$.

Thus, the transformation of the original expression results in the new expression:

$\left(3\times5\right)^{-3\times-6}$

Comparing with the provided choices, we see:

• Choice 1: $\left(3\times5\right)^{-3\times-6}$ - This matches our solution.

• Choices 2, 3, and 4 do not match the derived steps based on multiplying exponents.

Thus, the correct answer is choice 1: $\left(3\times5\right)^{-3\times-6}$.