Why Exponents Subtract

The exponents subtract because the factors cancel in pairs. To see this, write $\frac{x^{6}}{x^{2}}$ out in full: six factors of $x$ on top and two factors of $x$ on the bottom. The fraction $\frac{x}{x}$ equals 1 whenever $x$ is not zero. So each factor on the bottom pairs with one factor on top, and the pair becomes a factor of 1. Multiplying by 1 does not change the product.

Two pairs cancel, and four factors of $x$ are left on top, so $\frac{x^{6}}{x^{2}} = x^{4}$. The subtraction $6-2$ does not subtract the values of the powers. It counts the factors that are left after the pairing.

The same pairing explains two cases you will meet later. One is when the top and the bottom have the same number of factors. The other is when the bottom has more factors than the top. Both come in the chapter on zero and negative exponents.