Simplify (25×2)^16 ÷ (25×2)^5: Power Division Problem

To solve the given expression $\frac{\left(25\times2\right)^{16}}{\left(25\times2\right)^5}$, we need to apply the Power of a Quotient Rule for Exponents. This rule states that when you have the same base, you can subtract the exponent of the denominator from the exponent of the numerator. The general formula is:

• $\frac{a^m}{a^n} = a^{m-n}$

Here, the base $a$ is $25 \times 2$, the numerator's exponent $m$ is 16, and the denominator's exponent $n$ is 5.

Now, apply the Power of a Quotient Rule:

$\frac{\left(25\times2\right)^{16}}{\left(25\times2\right)^5} = \left(25\times2\right)^{16-5}$

Subtract the exponents:

$\left(25\times2\right)^{16-5} = \left(25\times2\right)^{11}$

Therefore, the simplified expression is:

$\left(25\times2\right)^{11}$

The solution to the question is: $\left(25\times2\right)^{11}$