Simplify (6×7)¹³ ÷ (7×6)¹⁹: Advanced Exponent Problem

To solve the problem $\frac{\left(6\times7\right)^{13}}{\left(7\times6\right)^{19}}$, we notice that the expressions in both the numerator and the denominator are very similar. Both involve the product of the numbers 6 and 7 raised to some power.

First, we can rewrite the denominator $\left(7 \times 6\right)^{19}$ as $\left(6 \times 7\right)^{19}$. This is possible because the multiplication is commutative, i.e., $a \times b = b \times a$.

Now, the expression becomes:

• $\frac{\left(6\times7\right)^{13}}{\left(6\times7\right)^{19}}$

We can use the rule of exponents, which states that when you divide like bases you subtract the exponents:

• $\frac{a^m}{a^n} = a^{m-n}$

Applying this rule to our expression, we have:

• $\frac{\left(6\times7\right)^{13}}{\left(6\times7\right)^{19}} = (6\times7)^{13-19}$
• $= (6\times7)^{-6}$

Next, we use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Therefore,

• $(6\times7)^{-6} = \frac{1}{(6\times7)^6}$

The solution to the question is: $\frac{1}{\left(6\times7\right)^6}$.