Why do we add exponents when we are multiplying like bases

Answer 1

When we multiply numbers with the same base, we add the exponents because of the fundamental rules of exponents.

The rule states that when you multiply two numbers with the same base, you add their exponents. This rule follows from the concept of repeated multiplication or the idea that taking a number to a power is equivalent to multiplying that number by itself.

For example, consider the expression 2³ * 2⁵. Here, both numbers have the same base (2), and we want to find their product. According to the rule of exponents, we add the exponents: 3 + 5 = 8. So, 2³ * 2⁵ is equal to 2⁸.

This rule makes sense because when you multiply like bases, you're essentially combining groups of the same number. For instance, 2³ represents 2 multiplied by itself three times, and 2⁵ represents 2 multiplied by itself five times. By multiplying them together, you're combining the groups of 2 into a larger group, which is represented by adding the exponents.