Final answer:
When multiplying numbers with the same base, exponents are added because the base is multiplied by itself an additional number of times for each exponent. In contrast, when raising an exponent to a power, the exponents are multiplied because it's equivalent to multiplying the base an additional number of times for each time it was already being multiplied by itself. Division of exponents with the same base results in subtracting the exponents.
Step-by-step explanation:
The rules for dealing with exponents vary depending on the operation being performed. When we multiply numbers with the same base and different exponents, such as am * an, we use the product law of exponents which tells us to add the exponents together to get am+n. This is because we are effectively multiplying the base a by itself m times and then n times, which is the same as multiplying it by itself m+n times. For example, 23 * 24 = 23+4 = 27. However, when we are applying an exponent to an entire expression with an exponent, such as (am)n, we use the power of a power law and multiply the exponents, resulting in amn. For example, (23)4 = 23*4 = 212.
Division of exponents, similarly to multiplication, will involve manipulation of the exponents. Dividing by a number with an exponent is equivalent to subtracting the exponent of the denominator from the exponent of the numerator if the bases are the same. So, am / an becomes am-n. For instance, 106 / 103 = 106-3 = 103.