Question

Why do you subtract exponents when dividing powers with the same base?

Answer 1

Let's see why that happens by evaluating an expression with exponents.

$\frac{a^{20}}{a^{10}}$ <-- Now expand the exponent
$\frac{a^{20}}{a^{10}} = \frac{(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)}{(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)}$

Now simplify the fraction by canceling out any common factors on the numerator and denominator.

$\frac{(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)}{(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)} = \frac{(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)}{1} = a^{10}$

So, it's the same thing as subtracting the powers.