Solve: (3×13)^14 ÷ (13×3)^20 Exponential Fraction

To solve the given problem, we need to carefully apply the rules of exponents, particularly the power of a quotient rule, which states $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$, and the rule for negative exponents, which is $a^{-m} = \frac{1}{a^m}$.

Given equation: $\frac{(3\times13)^{14}}{(13\times3)^{20}}$

First, notice that in both the numerator and the denominator, the terms are the same, just written in reverse order:

• Numerator: $(3\times13)^{14}$

• Denominator: $(13\times3)^{20}$

Since multiplication is commutative, we have:

• $(3\times13) = (13\times3)$

Therefore, the expression simplifies to:

• $\frac{(3\times13)^{14}}{(3\times13)^{20}}$

Since the bases are now identical, we can apply the rule $\frac{a^m}{a^n} = a^{m-n}$:

The exponent in the numerator is 14 and in the denominator is 20, giving us:

• $(3\times13)^{14-20}$

Calculate the subtraction in the exponent:

• $14 - 20 = -6$

Ultimately, the expression simplifies to:

• $(3\times13)^{-6}$

Therefore, the solution to the problem is: $(3\times13)^{-6}$