Exponential Notation
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 2^{12}.
Exponential Vocabulary
We use exponential notation to write repeated multiplication, such as 10 • 10 • 10 as 10^{3}. The 10 in 10^{3 }is called the base. The 3 in 10^{3 }is called the exponent. The expression 10^{3 }is called the exponential expression.
^{ }
^{ }base →^{ } 10^{3 }^{←exponent}
10^{3} is read as “10 to the third power” or “10 cubed.” It means 10 • 10 • 10, or 1,000.
8^{2} is read as “8 to the second power” or “8 squared.” It means 8 • 8, or 64.
5^{4} is read as “5 to the fourth power.” It means 5 • 5 • 5 • 5, or 625.
b^{5} 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 xy^{4}, only the y is affected by the 4. xy^{4} means x • y • y • y • y.
If the exponential expression is negative, such as −3^{4}, 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) = x^{4}, while −x^{4} = –(x • x • x • x).
You can see that there is quite a difference, so you have to be very careful!
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 5x^{3} if x = 4, first substitute the value 4 for the variable x. Then evaluate, using order of operations.
Example  
Problem  Evaluate. 5x^{3} if x = 4 

 5 • 4^{3}  Substitute 4 for the variable x. 
 5(4 • 4 • 4) = 5 • 64  Evaluate 4^{3}. 
 320  Multiply. 
Answer  5x^{3 }= 320 when x = 4 
Notice the difference between the example above and the one below.
Example  
Problem  Evaluate. (5x)^{3} if x = 4 

 (5 • 4)^{3}  Substitute 4 for the variable x. 
 20^{3}  Multiply. 
 20 • 20 • 20 = 8,000  Evaluate 20^{3}. 
Answer  (5x)^{3} = 8,000 when x = 4 
The addition of parentheses made quite a difference!
Example  
Problem  Evaluate. x^{3} if x = −4 

 (−4)^{3}  Substitute −4 for the variable x. 
 −4 · −4 · −4  Evaluate. 
 −4 · −4 · −4 = −64  Multiply. 
Answer  x^{3} = −64, when x = −4 
Evaluate the expression − (2x)^{4}, if x = 3.
A) 1,296 B) −1,296 C) 162 D) −162

What does it mean when an exponent is 0 or 1? Let’s consider 25^{1}. Any value raised to the power of 1 is just the value itself. This makes sense, because the exponent of 1 means the base is used as a factor only once. So the base stands alone, and 25^{1} is simply 25.
But what about a value raised to the power of 0? Use what you know about powers of 10 to find out what the power of 0 means. Below is a list of powers of 10 and their equivalent values. Look at how the numbers change going down the left and right columns. There’s a pattern there—see it?
Exponential Form  Expanded Form  Value 
10^{5}  10^{ }• 10 • 10 • 10 • 10  100,000 
10^{4}  10^{ }• 10 • 10 • 10  10,000 
10^{3}  10^{ }• 10 • 10  1,000 
10^{2}  10 • 10  100 
10^{1}  10  10 
Moving down the table, each row drops one factor of 10 from the one above it. From row 1 to row 2, the exponential form goes from 10^{5} to 10^{4}. The value drops from 100,000 to 10,000. Another way to put this is that each value is divided by 10 to produce the next value down the column.
Let’s use this pattern of division by 10 to predict the value of 10^{0}.
Exponential Form  Expanded Form  Value 
10^{5}  10^{ }• 10 • 10 • 10 • 10  100,000 
10^{4}  10^{ }• 10 • 10 • 10  10,000 
10^{3}  10^{ }• 10 • 10  1,000 
10^{2}  10 • 10  100 
10^{1}  10  10 
10^{0}  1  1 
Following the pattern, you see that 10^{0} is equal to 1. Would the pattern hold for a different base? Say a base of 3?
Exponential Form  Expanded Form  Value 
3^{5}  3^{ }• 3 • 3 • 3 • 3  243 
3^{4}  3^{ }• 3 • 3 • 3  81 
3^{3}  3 • 3 • 3  27 
3^{2}  3^{ }• 3  9 
3^{1}  3  3 
3^{0}  1  1 
Yes! And the same pattern would hold true for any nonzero number or variable raised to a power of 0, n^{0} = 1.
There is a conflict when the base is 0. You know that 0^{3} = 0, 0^{2} = 0, and 0^{1} = 0, so you would expect 0^{0} to also be equal to 0. However, the above pattern says that any base raised to the power of 0 is 1, so this leads you to believe that 0^{0} = 1. Notice the competing patterns—0^{0} cannot be both 0 and 1! In this case, mathematicians say that the value of 0^{0} is undefined. (And remember that undefined is not the same as 0!)
Exponents of 0 or 1
Any number or variable raised to a power of 1 is the number itself. n^{1} = n
Any nonzero number or variable raised to a power of 0 is equal to 1. n^{0} = 1
The quantity 0^{0} is undefined.

Example  
Problem  Evaluate. 2x^{0} if x = 9 

 2 • 9^{0}  Substitute 9 for the variable x. 
 2 • 1  Evaluate 9^{0}. 
 2  Multiply. 
Answer  2x^{0 }= 2, if x = 9 

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.
Evaluate the expression 3x^{0} – y^{1}, if x = 12 and y = –6.
A) 42 B) −3 C) 9 D) 2

What does it mean when an exponent is a negative integer? Let’s use the powers of 10 pattern from earlier to find out. If you continue this pattern to add some more rows, beyond 10^{0}, you find the following:
Exponential Form  Expanded Form  Value 
10^{5}  10^{ }• 10 • 10 • 10 • 10  100,000 
10^{4}  10^{ }• 10 • 10 • 10  10,000 
10^{3}  10^{ }• 10 • 10  1,000 
10^{2}  10 • 10  100 
10^{1}  10  10 
10^{0}  1  1 
10^{1}  _{}  _{} 
10^{2}  _{}
 _{} 
Following the pattern, you see that 10^{0} is equal to 1. Then you get into negative exponents: 10^{1} is equal to _{}, and 10^{2} is the same as _{}.
Following this pattern, a number with a negative exponent can be rewritten as the reciprocal of the original number, with a positive exponent.
For example, 10^{3 }=_{} and 10^{7}=_{}.
To see if these patterns hold true for numbers other than 10, check out this table with powers of 3.
Exponential Form  Expanded Form  Value 
3^{5}  3^{ }• 3 • 3 • 3 • 3  243 
3^{4}  3^{ }• 3 • 3 • 3  81 
3^{3}  3 • 3 • 3  27 
3^{2}  3^{ }• 3  9 
3^{1}  3  3 
3^{0}  1  1 
3^{}^{1}  _{}  _{} 
3^{}^{2}  _{}  _{} 
The numbers are different but the patterns are the same. We are now ready to state the definition of a negative exponent.
Negative Exponent
For any nonzero number n and any integer x, n^{x}= _{}. For example, 5^{2}= _{}.

Note that the definition above states that the base, n must be a “nonzero number.”
Evaluate the expression (x^{–2}) • (x^{0}) when x = 6.
A) _{} B) _{} C) 0 D) 36

Summary
Exponential notation is composed of a base and an exponent. It is a “shorthand” way of writing repeated multiplication, and indicates that the base is a factor and the exponent is the number of times the factor is used in the multiplication. The basic rules of exponents are as follows:
· An exponent applies only to the value to its immediate left.
· When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.
· For any nonzero number n, n^{0} = 1.
· For any nonzero number n and any integer x, n ^{–x }= _{}.