Simplify (15×4)³/(15×4)⁹: Working with Power Laws

Let's simplify the expression $\frac{\left(15\times4\right)^3}{\left(4\times15\right)^9}$:

We start by recognizing that both the numerator and the denominator share the same base: $(15 \times 4)$. Therefore, we have a quotient of powers with the same base:

$\frac{\left(15\times4\right)^3}{\left(15\times4\right)^9}$

According to the rules of exponents, when dividing like bases, we subtract the exponents:

$(15 \times 4)^{3 - 9}$

Subtracting the exponents, we have:

$(15 \times 4)^{-6}$

This matches with one of the choices:

• Choice 1: $(15\times4)^{3-9}$ is correct, as it simplifies to $(15\times4)^{-6}$.
• Choice 2 and Choice 3 involve incorrect operations on exponents (multiplication and addition respectively).
• Choice 4 reverses the subtraction order, which results differently in base powers than what is needed.

Therefore, the correct answer to the problem is:
$\left(15\times4\right)^{3-9}$.