Simplify (10×2)^20 ÷ (2×10)^7: Exponential Division Problem

To solve the problem, we first need to apply the exponent rules, specifically focusing on the "Power of a Quotient" rule. The given expression is:

$\frac{\left(10\times2\right)^{20}}{\left(2\times10\right)^7}$

We can notice that both the numerator and the denominator have the same base, which is $(10 \times 2) \ or \ (2 \times 10)$. Hence, let's simplify the base:

• $a = 10 \times 2 = 20$

Thus, both the numerator and the denominator can be rewritten with the base $a$:

• $a^{20}$ for the numerator

• $a^{7}$ for the denominator

Now, using the "Power of a Quotient" rule:

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

We apply this rule to our expression:

$\frac{a^{20}}{a^7} = a^{20-7}$

This simplifies to:

$a^{13}$

Substituting back the value of $a$:

$\left(2 \times 10\right)^{13}$

However, let's check the solution form given in the problem:

The solution hinted at is:

$\left(2 \times 10\right)^{20-7}$

Indeed, it verifies our calculation that the expression simplifies to $\left(2 \times 10\right)^{13}$.

The solution to the question is: $\left(2 \times 10\right)^{13}$