Simplify (4×3)⁶/(3×4)⁹: Complex Exponential Expression

The given expression is: $\frac{\left(4\times3\right)^6}{\left(3\times4\right)^9}$.

We want to simplify this expression using the laws of exponents.

First, notice that the base is the same in both the numerator and the denominator: $4 \times 3$. We can apply the property of exponents that involves a quotient:

$\frac{a^m}{a^n} = a^{m-n}$

Thus, we can rewrite the expression as:

$\left(4 \times 3\right)^{6-9}$

Subtract the exponents: $6 - 9 = -3$.

The expression now becomes:

$\left(4 \times 3\right)^{-3}$

To express this with a positive exponent, recall the rule:

$a^{-n} = \frac{1}{a^n}$

Therefore, $\left(4 \times 3\right)^{-3}$ can be written as:

$\frac{1}{\left(4 \times 3\right)^3}$

The solution to the question is: $\frac{1}{\left(4\times3\right)^3}$