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Section 1.7 Exponents and Order of Operations
Basic College Mathematics, Miller/O’Neill/Hyde
Andrea Hendricks
I. Connection
In section 1.5, we learned that multiplication is repeated addition. In this section, we will
learn how to represent repeated multiplication.
II. Overview – In this section, you will learn about
A. Exponents
B. Square Roots
C. Order of Operations
D. Computing a Mean (Average)
III. Exponents
A. Exponent or power – a number that represents how many times a factor is repeated
in a product.
1) The product 2  2  2 can be written as 23
2) The number 2 is called the base and it represents the number that is being
repeated in the product..
3) The number 3 is called the exponent and it indicates how many times the base,
2, is being multiplied by itself.
B. Special Exponents and Bases
1) A number raised to the first power always results in the number itself. A
number written without a power is assumed to be raised to the first power.
2) Squares
a) Squares of whole numbers are used in many applications.
b) It would benefit you to learn some of these perfect squares.
c) A good rule of thumb is to know by memory the numbers
12 , 22 ,32 ,...152
3) Powers of 10
a) Powers of 10, that is, 10 raised to a whole-number power, are important
numbers in our number system.
b) You should know and recognize the pattern that results with these
expressions.
101  10, 102  100, 103  1000 and so on.
c) The number of zeros is the same as the exponent on the base of 10.
IV. Square Roots
A. Finding a square root is the reverse process of squaring.
B. The symbol
, called a radical sign, denotes the positive square root of a number.
C. To find a positive square root of a number, we have to find the number that must be
squared to get the given number.
Example: To find 49 , we must find what positive number squared, equals 49.
Since 7 2  49, 49  7 .
D. Knowing the perfect squares will make the process of finding the square roots
much easier.
V. Order of Operations
A. The order of operations is the accepted guidelines for simplifying numerical
expressions that contain more than one operation.
B. Illustrate why these guidelines are needed.
1) Compare the two possible results of 6 – 2(3).
2) One possible result is 6 – 2(3) = 4(3) = 12.
3) Another result could be 6 – 2(3) = 6 – 6 = 0.
4) Without guidelines, there would be a lot of confusion in the study of
mathematics.
C. The correct result is the 0 because of the priority of operations.
D. Order of Operations
1) Perform all operations inside parentheses first.
2) Simplify any expressions containing exponents or square roots.
3) Perform multiplication or division in the order that they appear from left to
right.
4) Perform addition or subtraction in the order that they appear from left to right.
VI. Computing a Mean (Average)
A. Definition – The mean of a group of numbers is the average of the numbers.
B. To calculate the mean, divide the sum of the values in the group by the number of
values in the group.
VII. Summary – You should be able to
A. Write a repeated multiplication using an exponent.
B. Expand an exponential expression as repeated multiplication.
C. Simplify exponential expressions.
D. Find square roots.
E. Simplify a numerical expression using the order of operations.
F. Compute the mean of a set of numbers.