Solve Expression: ((5×3)^4)^(-3) Using Power Rules

To solve the problem, let us simplify the expression $\left(\left(5\times3\right)^4\right)^{-3}$.

First, recognize that the expression inside the parentheses, $5 \times 3$, can be multiplied to give us 15. However, we'll focus on exponent rules directly.

• Step 1: Apply the Power of a Power Rule.
  Using the formula $(a^m)^n = a^{m \times n}$, simplification gives: $\left(\left(5\times3\right)^4\right)^{-3} = \left( (5 \times 3)^{4 \times -3} \right) = \left(5\times3\right)^{-12}.$
• Step 2: Apply the Negative Exponent Rule.
  Using $a^{-n} = \frac{1}{a^n}$, the expression becomes: $\left(5\times3\right)^{-12} = \frac{1}{\left(5\times3\right)^{12}}.$

Therefore, the solution to the given expression is $\frac{1}{\left(5\times3\right)^{12}}$.

Now, let's verify the answer with the choices provided:

• Choice 1: $\frac{1}{\left(5\times3\right)^{12}}$ - This matches our solution.
• Choice 2: $(5\times3)^{12}$ - Incorrect, doesn't account for the negative exponent.
• Choice 3: $\frac{1}{\left(5\times3\right)^{-1}}$ - Incorrect power and doesn't represent the full expression.
• Choice 4: $\frac{1}{\left(5\times3\right)^1}$ - Incorrect power and doesn't reflect the original problem.

Thus, the correct choice is Choice 1: $\frac{1}{\left(5\times3\right)^{12}}$.