Simplify (5×2)^8 ÷ (2×5): Power Reduction Problem

Let's break down the expression and apply the rules of exponents step by step. We start with the given expression:

$\frac{\left(5\times2\right)^8}{\left(2\times5\right)^{} }$

Step 1: Simplify the denominator

The denominator $\left(2\times5\right)$ is equivalent to $10$, but because it does not have an exponent specified, it is effectively raised to the power of 1:

$\left(2\times5\right)^1 = 10^1 = 10$

Step 2: Apply the power of a quotient rule

The expression $\frac{\left(5\times2\right)^8}{\left(2\times5\right)}$ can be simplified by applying the power of a quotient rule for exponents:

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

Here $a = \left(5 \times 2\right)$, $m = 8$, and the exponent in the denominator is $n = 1$ because$(2 \times 5) = 10^1$. Applying the quotient rule gives:

$\left(5\times2\right)^{8-1}$

This simplifies to:

$\left(5\times2\right)^7$

Final Answer

The simplification results in:

$\left(5\times2\right)^7$