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Chapter 1

Introduction to Algebraic

Expressions

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Exponential Notation and Order of Operations

• Exponential Notation

• Order of Operations

• Simplifying and the Distributive Law

• The Opposite of a Sum

1.8

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Exponential Notation

• The 5 is called an exponent.

• The 4 is the base.

factors

5

5

4 4 4 4 4 we write as 4

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Example

Write exponential notation for 777777.

Solution Exponential notation is 76

7 is the base.

6 is the exponent.

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Example

Evaluate a) 82 b) (−8)3 c) (4y)3

Solutiona) 82 = 8 8 = 64

b) (−8)3 = (−8) (−8) (−8) = 64(−8) = − 512

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Example

c) (4y)3 = (4y) (4y) (4y) = 4 4 4 y y y

= 64y3

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Exponential Notation For any natural number n,

factors

means ... .n

nb b b b b b

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Rules for Order of Operations1. Simplify, if possible, within the innermost

grouping symbols, ( ), [ ], { }, | |, and above or below any fraction bars.

2. Simplify all exponential expressions.

3. Perform all multiplications and divisions, working from left to right.

4. Perform all additions and subtractions, working from left to right.

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Example

Simplify:

Solution

20 4 2 7

20 4 2 7 20 8 7

712

19

Multiplying

Subtracting and adding from left to right

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Example

Evaluate 16 8(7 y)2 for y = 2.

Solution16 8(7 y)2 = 16 8(7 2)2

= 16 8(5)2 = 16 8(25) = 2(25)

= 50

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Example

Simplify:

Solution

2 320 5 4 [(13 4) 8] 2 .

2 320 5 4 [(13 4) 8] 2 4 16[17 8] 8

4 16(9) 8

4 144 8

140

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Example

Simplify:

2

4(6 2) 8(8 3)6(4 2) 2

2

4(6 2) 8(8 3) 4(8) 8(5)6(2) 46(4 2) 2

32 4012 4

72 98

Solution

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Example

Simplify:

Solution

7 3 4(2 5)x x

7 3 (2 5)4 07 8 23x x x x

15 23x

Distributive Law

Combining Like Terms

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Example

Write an expression equivalent to (4x + 3y + 5) without using parentheses.

Solution

(4x + 3y + 5) = 1(4x + 3y + 5)

= 1(4x) + 1(3y) + 1(5)

= 4x 3y 5

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The Opposite of a Sum For any real numbers a and b,

−(a + b) = −a + (−b) = −a − b

(The opposite of a sum is the sum of the opposites.)

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Example

Simplify:

Solution

2 28 3 (2 6 )y y y y

2 2 2 2( )8 3 2 6 8 3 2 6y y y y y y y y

26 3y y

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Example

Simplify:

Solution

3 37 5 [3( 2) 1]w w

3 33 37 5 [3( 2) 1] 7 5 [ 13 6 ]w w ww 3 37 5 [3 5]w w 3 37 5 3 5w w 34 10w