Why do we subtract exponents when we are dividing like bases?
When we divide two numbers with the same base, we subtract the exponents because of the basic properties of exponents and the rules of division.
When we multiply two numbers with the same base, the exponents are added together:
a^m * a^n = a^(m+n)
For example, 2^3 * 2^2 = 2^(3+2) = 2^5.
Now, consider dividing two numbers with the same base:
a^m / a^n = a^(m-n)
This can be derived from the fact that division is the inverse operation of multiplication. If we divide a number by another, we are essentially multiplying with the reciprocal of the divisor:
a^m / a^n = a^m * (1/a^n)
Using the properties of exponents, we can rewrite 1/a^n as a^(-n):
a^m * a^(-n) = a^(m + (-n))
Adding -n is the same as subtracting n, so we end up with:
a^m / a^n = a^(m - n)
Therefore, when we divide two numbers with the same base, we subtract the exponents to simplify the expression.