Why do we subtract exponents when we are dividing like bases?

Answer 1

When we divide two numbers with the same base, we manipulate the exponents in order to simplify the expression.

To see why we subtract exponents, let's consider an example:

Let's divide \(a^m\) by \(a^n\), where \(m\) is greater than or equal to \(n\).

Using the definition of division, \(a^m\) divided by \(a^n\) is the same as \(a^m \times a^{-n}\).

Now, using the rules of exponents, we know that when we multiply two numbers with the same base, we add their exponents.

So, \(a^m \times a^{-n}\) becomes \(a^{m+(-n)}\), which is \(a^{m-n}\).

Therefore, when we divide like bases, we subtract the exponents.