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Order of operation and basic properties of exponents
SM1 Section 1-2
Date _______
Objective: Review order of operation and basic properties of exponents
The following calculation could have several different results depending upon which order we complete
the operations. 8  3 6  4  2
For each of these examples there is a space to the right provided for you to write down what operation
was completed.
For example, we could just work from left to right doing each operation in order to get:
8  3 6  4  2  31
We could work from right to left, doing each operation as they appear in this order
8  3 6  4  2  20
We could do some of the operations in the middle first and then the outside operations to get:
8  3 6  4  2  15
We could also perform the operations in another order to get
8  3 6  4  2  24
Notice that we have calculated these numbers and operations in four different ways and we got different
answers each time. This is confusing and leads to miscommunication when working with math. To
avoid this confusion, we have a predetermined order that must be followed so that everyone will get the
same results when performing the calculations.
Because multiplication is multiple addition, 3  4 means we have 3 fours added together, 2  3  4 would
mean 2  4  4  4 this set of operations are all the same and we get a sum of 14. This is the only correct
answer for this calculation. But if we add 2 + 3 first and get 5 and then multiply by 4 we get 20. We
must do the multiplication first in order to get the right answer.
The order of operations has been set in order to help us get the correct calculation each time.
Order of operations
1st
Grouping symbols (i.e. Parentheses, brackets)
nd
2
Exponents
3rd
Multiplication and division
th
4
Addition and subtraction
You may also remember PEMDAS (Please Excuse My Dear Aunt Sally) meaning Parentheses,
Exponent, Multiplication & Division, Addition & Subtraction.
If we want to do a calculation out of order, then we must identify it with parenthesis or other grouping
symbols.
For example
6  4  2 requires us to divide 4 by 2 first to get 6 + 2 or 8
(6  4)  2 requires us to look at the 6 + 4 as a number that will be divided by 2 it can be written as
64
in this case we get 10 divided by 2 which is 5.
2
Calculate the following using the order of operations
52  2  32  2
32  72  22  9  2
8  32  4  2  9  82  8
One way to practice this order of operations is to play a game called crypto. In this game, you are given
a target number from 11 – 66 and 5 working numbers 1 – 6. You are to use all the working numbers
using any operations to arrive at the target number.
Example 1. If we are given the target number of 14 and the working numbers 2, 3, 3, 6, 4. We can get
the target number by 6 + 4 + 3 + 3 – 2. Since the only operations used are addition and subtraction, we
do not care about the order. However, there are times in the game when we will have to write the
solution in an algebraic expression using grouping symbols in order to get the target number.
Example 2. Target number is 17, working numbers are 1, 2, 5, 6, 3
6 – 2 = 4 and 4 times 3 = 12 and 12 + 5 = 17 and 17 times 1 equals 17. The algebraic expression
representing this calculation is   6  2  3  5 1 and the order of operations guarantees the answer.
You can generate the target number by rolling two dice, one for the tens and one for the ones. Then
obtain the working numbers by rolling five dice.
Example 3.
The gray die will be our tens and the white die will be our ones so the roll shown give a target
number of 41. The working numbers are 5, 3, 5, 3, 2
A solution would be the algebraic expression :  5  3  5  3  2
Play Crypto with the following numbers. Write your algebraic expressions on this page.
Game 1
Game 2
Game 3
Game 4
Properties of Exponents
If x3  x  x  x and x 4  x  x  x  x then x 3  x 4  __________________________ or _____
Product of Powers
For all real numbers x and all integers m and n, x m  x n  x m  n
If x 6  x  x  x  x  x  x and x3  x  x  x then x6  x3 
x6
or ____________________ or _____
x3
Quotient of Powers
For all real numbers x (except zero) and all integers m and n,
If x 6  x  x  x  x  x  x then x6  x6 
x6
x6
xm
 x mn
n
x
or ____________________________ or ______
Zero Property of Exponents
For all real numbers x (except zero), x 0  1
x3
If x  x  x  x  x  x  x and x  x  x  x then x  x  6 or _____________________ or ______
x
Negative Exponents
1
For all real numbers x (except zero) and all integers n, x  n  n
x
6
3
3
6
If x 7  x  x  x  x  x  x  x then  x 7   ___________________________________ or ______
2
Power of a power
For all real numbers x and all integers m and n,  x m   x mn
n
If ( xy)3   xy  xy  xy  or _____________________________ or ________
Power of a Product
For all real numbers x and y, and all integers n
 xy 
n
 xn y n
3
 x   x  x  x 
If        or
 y   y  y  y 
____________________________________ or ________
Power of a Quotient
n
x
xn
For all real numbers x and y (except 0), and all integers n    n
y
 y
We discovered earlier in the year that the distributive property is true for all real numbers. This means
1
we can simplify an expression such as 3  x 6  x 4  where x  0 .
x
1 6
1
1
x  x4   3  x6  3  x4
3 
x
x
x
6
1 6 1 4 x
x4

x


x


x3
x3
x3 x3
x6 x4
 3  x3  x
3
x
x
3
x  x  x4
Practice:
Simplify each expression: (Show your work).
 3d 
b 2  b5
2 x(5 x  6 x  x  7)
3
 w3 
 3
w 
2
4
 3d 
3
  5x
 4k 2 


 3 

2 4
 13 x3 yz 2 

5 2 
 22 x y z 
n
m
Explain why   is the same as
n
n
 
m
3
n
0
3
3x y 
4
2 2
4 f 2 2 f 3
24m4 n
 2mn 
3
 2d 
2 3