Final answer:
In mathematics, when multiplying exponentials, multiply the bases and add the exponents; for division, divide the bases and subtract the exponents. This results in easier handling of large numbers, simplifying calculations.
Step-by-step explanation:
Understanding Multiplication and Division of Exponents
When it comes to multiplying exponentials, the process is straightforward: you multiply the base numbers normally and add the exponents if the bases are the same. For example, when multiplying 3.2 × 10³ and 2 × 10², you would first multiply 3.2 by 2 to get 6.4, and then add the exponents (3 and 2) to get 10³×10² = 10⁵, resulting in 6.4 × 10⁵.
For division, the rule is similar but the exponents are subtracted instead of added. So, if we were to divide 10⁶ by 10³, we would subtract the exponents (6 - 3) to get 10³. Likewise, if dividing 4.5 × 10⁹ by 2 × 10³, you would divide 4.5 by 2 and subtract 3 from 9, which results in 2.25 × 10⁶. This simplification makes working with large numbers less cumbersome.
When exponents of the same base are multiplied, we multiply the exponents. For example, (5³)⁴ is equal to 5¹² because you multiply the exponents 3 and 4 to get 12. Thus, exponential terms are significantly simplified by applying the rules of exponent multiplication and division.