Why do we add exponents when we are multiplying like bases?

Answer 1

When we multiply two numbers with the same base, we add the exponents because of the properties of exponents and the idea of repeated multiplication.

Let's consider an example to illustrate this concept: 2^3 * 2^2.

The base in both terms is 2. To understand why the exponents are added, we can break down the expression using the concept of repeated multiplication.

2^3 represents 2 * 2 * 2, which equals 8.
2^2 represents 2 * 2, which equals 4.

When we multiply these two terms together, we can think of it as multiplying 8 by 4: 8 * 4 = 32.

Now let's see how this aligns with adding the exponents:
2^3 * 2^2 = 2 * 2 * 2 * 2 * 2 = 2^5.
The exponent 5 represents the fact that we have five 2's multiplied together.

So, by adding the exponents, we can express the multiplication of like bases more efficiently. In this example, instead of writing out the repeated multiplication of twos, we simply add the exponents to represent the same calculation.