ch 1 sec 8: slide #1 columbus state community college chapter 1 section 8 exponents and order of...

Ch 1 Sec 8: Slide #1

Columbus State Community College

Chapter 1 Section 8

Exponents and Order of Operations



1. Use exponents to write repeated factors.

2. Simplify expressions containing exponents.

3. Use the order of operations.

4. Simplify expressions with fraction bars.


Exponents

An exponent is a quick way to write repeated multiplication.

For example,

Base

3 • 3 • 3 • 3 can be written 34 Exponent

This is called exponential notation or exponential form.

To simplify 34, actually do the multiplication.

34 = 3 • 3 • 3 • 3 = 81


Reading Common Exponents

Here are some examples of how to read common exponents.

31 is read “3 to the first power.”

32 is read “3 to the second power” or, more commonly, “3 squared.”

33 is read “3 to the third power” or, more commonly, “3 cubed.”

34 is read “3 to the fourth power.”

35 is read “3 to the fifth power.”


Using Exponents

EXAMPLE 1 Using Exponents

Rewrite each multiplication using exponents. Also indicate how to read the exponential form.

(a) 10 • 10 • 10 • 10 • 10 • 10 can be written as 106, which is read “10 to the sixth power.”

(b) ( 7 )( 7 ) can be written as 72, which is read “7 squared” or “7 to the second power.”

(c) 2 can be written as 21, which is read “2 to the first power.”


Using Exponents with Negative Numbers

EXAMPLE 2 Using Exponents with Negative Numbers

Simplify.

(a) ( –2 )3 = ( –2 ) ( –2 ) ( –2 )

4 ( –2 )

–8

(b) ( –2 )4 = ( –2 ) ( –2 ) ( –2 ) ( –2 )

–8 ( –2 ) From 2(a), ( –2 )3 = –8.

16


Using Exponents with Negative Numbers

EXAMPLE 2 Using Exponents with Negative Numbers

Simplify.

(c) ( –5 )2 ( –2 )3

( 25 ) ( –8 )

–200

= ( –5 ) ( –5 ) ( –2 ) ( –2 ) ( –2 )

( –5 )2 ( –2 )3


^

Calculator Tip – TI-30X IIS

Calculator Tip On your TI-30X IIS calculator, use the exponent key to enter exponents. To enter 75, press the following keys.

^

7 ^ 5 = 16807


Working from Left to Right

EXAMPLE 3 Working from Left to Right

Simplify.

(a) –3 – –8 + –2 Do additions and subtractions from left to right.

5 + –2

3


Working from Left to Right

EXAMPLE 3 Working from Left to Right

Simplify.

(b) –20 ÷ 2 • 5 Do multiplications and divisions from left to right.

–10 • 5

–50


Mixing Operations

10 + 2 • 3

12 • 3

36

10 + 2 • 3

10 + 6

16

Compare the methods used to simplify the following example.

If we work from left to right If we multiply first

10 + 2 • 3

Mathematicians have agreed to do things in a certain order. In this example, we multiply before we add.


Order of Operations

Order of Operations

Step 1 Work inside parentheses or other grouping symbols.

Step 2 Simplify expressions with exponents.

Step 3 Do the remaining multiplications and divisions as they occur from left to right.

Step 4 Do the remaining additions and subtractions as they occur from left to right.


CAUTION

CAUTIONTo help in remembering the order of operations, you may have memorized the letters PEMDAS, or the phrase “Please Excuse My Dear Aunt Sally.”

Please Excuse My Dear Aunt Sally

1. Parentheses 2. Exponents 3. Multiply & Divide (from left to right) 4. Add & Subtract (from left to right)

Be careful! Do not automatically do all multiplication before division. Multiplication and division are done from left to right. Likewise, addition and subtraction are done from left to right.


Calculator Tip

Calculator Tip Enter the previous example in your calculator.

10 3 = 16+ x2

If you have a scientific calculator, it automatically uses the order of operations and multiplies first to get the correct answer of 16.


Using the Order of Operations with Whole Numbers

EXAMPLE 4 Using the Order of Operations with Whole #’s

Simplify.

5 + 2 ( 24 – 4 • 2 ) ÷ 8 Multiply inside parentheses first.

5 + 2 ( 24 – 8 ) ÷ 8 Subtract inside parentheses.

5 + 2 ( 16 ) ÷ 8

5 + 32 ÷ 8

5 + 4

9

Multiply.

Divide.

Add.


Using the Order of Operations with Integers

EXAMPLE 5 Using the Order of Operations with Integers

Simplify.

–40 ÷ ( 12 – 7 ) – 4 Subtract inside parentheses first.

–40 ÷ 5 – 4 Divide.

–8 – 4

–12

Subtract.

(a)


Using the Order of Operations with Integers

EXAMPLE 5 Using the Order of Operations with Integers

Simplify.

(b) 6 + 5 ( 1 – –3 ) • ( 18 ÷ –9 ) Subtract inside parentheses first.

6 + 5 ( 4 ) • ( 18 ÷ –9 ) Divide inside parentheses.

6 + 5 ( 4 ) • ( –2 )

6 + 20 • ( –2 )

6 + –40

–34

Multiply from left to right.

Multiply.

Add.


Using the Order of Operations with Exponents

EXAMPLE 6 Using the Order of Operations with Exponents

Simplify.

( –5 )2 – ( –2 )3 Apply exponents.

25 – –8 Add.

33

(a)


Using the Order of Operations with Exponents

EXAMPLE 6 Using the Order of Operations with Exponents

Simplify.

(b) ( –3 )3 – ( 8 – 6 )3 ( 3 )2 Subtract inside parentheses first.

( –3 )3 – ( 2 )3 ( 3 )2 Apply exponents.

–27 – 8 ( 9 )

–27 – 72

–99

Multiply.

Add.


Using the Order of Operations with Fraction Bars

EXAMPLE 7 Using the Order of Operations with Fraction Bars

Simplify.

First, do the work in the numerator

–4 – 2 ( 3 – –1 )2

–4 – 2 ( 3 – –1 )2

3 • –4 ÷ 2 • –3

–4 – 2 ( 4 )2

–4 – 2 ( 16 )

–4 – 32

–36

Next, do the work in the denominator

3 • –4 ÷ 2 • –3

–12 ÷ 2 • –3

–6 • –3

18

–3618

= = –2



Chapter 1 Section 8 – End

Written by John T. Wallace

ch 1 sec 8: slide #1 columbus state community college chapter 1 section 8 exponents and order of...

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