Exponents and Order of Operations
1. Exponents:
An exponent is a number that indicates how many
times the base is to be used as a factor. Exponents indicate
repeated multiplication.
The base is the number that is multiplied.
Ex: Identify the base and the exponent: 2^{
3}
Two is the base and three is the exponent.
2. Zero exponent:
Any number other than 0 raised to the zero power
is 1.
3. Order of operations:
When evaluating mathematical expressions,
we will perform operations in the following order:
First: If the expression contains grouping symbols, such as
parenthesis (), brackets [ ], braces { }, or a fraction bar, then
we perform the operations inside the grouping symbols, or
above and below the fraction bar, first.
Second: Evaluate, or simplify, any numbers with exponents.
Third: Do all multiplications and divisions in order from left to
right.
Fourth: Do all additions and subtractions in order from left to
right.
Ex: Simplify.
2 + 3â€¢5
2 + 3â€¢5 

(operations of addition and multiplication are present)


= 2 + 15 
(perform multiplication first)


= 17 
(perform addition second) 
b. 3 + 2â€¢4^{2}
3 + 2â€¢4^{2} 

(operations of addition, mult, and exponentiation) 

= 3 + 2â€¢16 
(do exponents first) 

= 3 + 32 
(do multplication second) 

= 35 
(do addition third) 
4. Vocabulary:
We will now translate into mathematical symbols
English phrases that contain complicated expressions involving
the terms sum, product, difference, and quotient.
English phrase 
Math symbols 
sum of a and b 
a + b 
two times the sum of a and b 
2(a + b) 
product of p and q 
p Â· q 
product of p and the sum of a and b 
p Â· (a + b) 
sum of p and the product of a and
b 
p + a Â· b 
difference of p and the sum of a
and b 
p  (a + b) 
sum of the product of a and b and
the product of c and d 
a Â· b + c
Â· d 
Ex: Translate each phrase into math symbols.
a. product of 4 and the sum of 3 and x
Answer: 4 Â· (3 + x)
b. difference of 4 and the sum of 3 and x
Answer: 4  (3 + x)
c. sum of the product of 4 and 3 and the product of 2 and
5
Answer: 4 Â· 3 + 2
Â· 5
d. twice the product of 4 and x
Answer: 2 Â· (4
Â· x)
e. twice the sum of 8 and 5
2 Â· (8 + 5)
