When dividing powers that have the same base, we subtract the exponent of the denominator from the exponent of the numerator. This might seem confusing at first, but there’s a logical reason behind this rule.
Let’s break it down with an example. Imagine you have the expression am / an. According to the rule we just mentioned, the result can be calculated as follows:
- We take the base ‘a’ and keep it the same.
- Then we subtract the exponent of the denominator (n) from the exponent of the numerator (m).
This leads us to the conclusion that: am / an = am-n.
To understand why this works, let’s look at another angle. The operation of division can be thought of as finding out how many times the denominator can fit into the numerator. When we have am, it means we have ‘a’ multiplied by itself ‘m’ times. When we divide by an, we are essentially removing ‘n’ occurrences of ‘a’ from ‘m’ occurrences of ‘a’. Thus, you are left with am-n.
This principle helps simplify many algebraic expressions and is foundational in manipulating powers and exponents. By using this subtractive method, we maintain consistency in how we handle these mathematical operations, making calculations involving exponents much more manageable.