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