Step-by-step explanation:
Learning Objective(s)
· Use the product rule to multiply exponential expressions with like bases.
· Use the power rule to raise powers to powers.
· Use the quotient rule to divide exponential expressions with like bases.
· Simplify expressions using a combination of the properties.
Introduction
Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier to leave the expressions in exponential notation when multiplying or dividing. Let’s look at rules that will allow you to do this.
The Product Rule for Exponents
Recall that exponents are a way of representing repeated multiplication. For example, the notation 54 can be expanded and written as 5 • 5 • 5 • 5, or 625. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.
What happens if you multiply two numbers in exponential form with the same base? Consider the expression (23)(24). Expanding each exponent, this can be rewritten as (2 • 2 • 2) (2 • 2 • 2 • 2) or 2 • 2 • 2 • 2 • 2 • 2 • 2. In exponential form, you would write the product as 27. Notice, 7 is the sum of the original two exponents, 3 and 4.
What about (x2)(x6)? This can be written as (x • x)(x • x • x • x • x • x) = x • x • x • x • x • x • x • x or x8. And, once again, 8 is the sum of the original two exponents.
The Product Rule for Exponents
For any number x and any integers a and b, (xa)(xb) = xa+b.
To multiply exponential terms with the same base, simply add the exponents.
