Answer:
Learning Objective(s)
· Evaluate expressions containing exponents.
· Evaluate exponential notations with exponents of 0 and 1.
· Write an exponential expression involving negative exponents with positive exponents.
Introduction
A common language is needed in order to communicate mathematical ideas clearly and efficiently. Exponential notation is one example. It was developed to write repeated multiplication more efficiently. For example, growth occurs in living organisms by the division of cells. One type of cell divides 2 times in an hour. So in 12 hours, the cell will divide 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 times. This can be written more efficiently as 212.
Exponential Vocabulary
We use exponential notation to write repeated multiplication, such as 10 • 10 • 10 as 103. The 10 in 103 is called the base. The 3 in 103 is called the exponent. The expression 103 is called the exponential expression.
base → 103 ←exponent
103 is read as “10 to the third power” or “10 cubed.” It means 10 • 10 • 10, or 1,000.
82 is read as “8 to the second power” or “8 squared.” It means 8 • 8, or 64.
54 is read as “5 to the fourth power.” It means 5 • 5 • 5 • 5, or 625.
b5 is read as “ b to the fifth power.” It means b • b • b • b • b. Its value will depend on the value of b.
The exponent applies only to the number that it is next to. So in the expression xy4, only the y is affected by the 4. xy4 means x • y • y • y • y.
If the exponential expression is negative, such as −34, it means –(3 • 3 • 3 • 3) or −81.
If −3 is to be the base, it must be written as (−3)4, which means −3 • −3 • −3 • −3, or 81.
Likewise, (−x)4 = (−x) • (−x) • (−x) • (−x) = x4, while −x4 = –(x • x • x • x).
You can see that there is quite a difference, so you have to be very careful!
Evaluating Expressions Containing Exponents
Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.
You can use PEMDAS to remember the order in which you should evaluate the expression. First, evaluate anything in Parentheses or grouping symbols. Next, look for Exponents, followed by Multiplication and Division (reading from left to right), and lastly, Addition and Subtraction (again, reading from left to right).
So, when you evaluate the expression 5x3 if x = 4, first substitute the value 4 for the variable x. Then evaluate, using order of operations.
Step-by-step explanation:
example: Problem
Evaluate.
5x3 if x = 4
5 • 43
Substitute 4 for the variable x.
5(4 • 4 • 4) = 5 • 64
Evaluate 43.
320
Multiply.
Answer
5x3 = 320 when x = 4
