The problem asks whether Neil made a mistake in simplifying the expression $\frac{5^{-6}}{5^{-4}}$. The steps Neil took are: Step 1: $\frac{5^{-6}}{5^{-4}} = 5^{-6-(-4)}$ Step 2: $5^{-6-(-4)} = 5^{-2}$ Step 3: $5^{-2} = \frac{1}{5^2}$

1. Problem Description

The problem asks whether Neil made a mistake in simplifying the expression $\frac{5^{-6}}{5^{-4}}$. The steps Neil took are:

Step 1: $\frac{5^{-6}}{5^{-4}} = 5^{-6-(-4)}$

Step 2: $5^{-6-(-4)} = 5^{-2}$

Step 3: $5^{-2} = \frac{1}{5^2}$

2. Solution Steps

To determine if Neil made a mistake, we need to simplify the original expression correctly and compare it to Neil's steps.

The rule for dividing exponents with the same base is:

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

Applying this rule to the expression:

$\frac{5^{-6}}{5^{-4}} = 5^{-6 - (-4)}$

Step 1: $5^{-6 - (-4)} = 5^{-6 + 4} = 5^{-2}$.

So far, Neil is correct.

Step 2: Neil states $5^{-6-(-4)} = 5^{-2}$.

$-6-(-4) = -6+4 = -2$. This is also correct.

Step 3: Neil states $5^{-2} = \frac{1}{5^2}$.

We know that $a^{-n} = \frac{1}{a^n}$. Therefore $5^{-2} = \frac{1}{5^2}$. This is also correct.

Since all the steps are correct, Neil did not make a mistake.

3. Final Answer

Neil did not make a mistake.