Simplify (ax)^21 / (xa)^6: Advanced Exponent Rules Practice

We start with the given expression:
$\frac{\left(a\times x\right)^{21}}{\left(x\times a\right)^6}$.

Firstly, notice that the terms inside the parentheses in both the numerator and the denominator are the same: $a \times x$ and $x \times a$, which are equivalent due to the commutative property of multiplication.

Thus, we can rewrite the expression in terms of $(ax)$ as follows:
$\frac{\left(a x\right)^{21}}{\left(a x\right)^6}$.

We will apply the power of a quotient rule for exponents, which states that:

• $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$.

Using this rule to simplify the expression, we have:

$\frac{(ax)^{21}}{(ax)^6} = (ax)^{21-6} = (ax)^{15}$.

Thus, the expression simplifies to $(a \times x)^{15}$.

Therefore, the solution to the question is:
$\left(x\times a\right)^{15}$.