Simplify (a×b)¹² ÷ (a×b)³: Power Division Problem

To solve the problem $\frac{\left(a \times b\right)^{12}}{\left(a \times b\right)^3}$, we can use the rule for exponents known as the Power of a Quotient Rule, which states that $\frac{x^m}{x^n} = x^{m-n}$, where $x$ is a non-zero base and $m$ and $n$ are the exponents.

Let's apply this rule step by step to our expression:

• Identify the base: In the expression $\frac{\left(a \times b\right)^{12}}{\left(a \times b\right)^3}$, the base is $a \times b$.
• Identify the exponents: The exponent for the numerator is 12, and for the denominator, it is 3.
• Apply the Power of a Quotient Rule: $\frac{\left(a \times b\right)^{12}}{\left(a \times b\right)^3} = \left(a \times b\right)^{12-3}$.

Thus, the simplification of the given expression is: $\left(a \times b\right)^{12-3}$

The solution to the question is: $\left(a \times b\right)^{9}$