1.3 1.7

Copyright © 2011 Pearson Education, Inc.

Adding and Subtracting Real Numbers; Properties of Real Numbers1.31.3

1. Add integers.2. Add rational numbers.3. Find the additive inverse of a number.4. Subtract rational numbers.

Slide 1- 2Copyright © 2011 Pearson Education, Inc.

Objective 1

Add integers.


Parts of an addition statement: The numbers added are called addends and the answer is called a sum.

2 + 3 = 5

Addends Sum


Adding Numbers with the Same Sign

To add two numbers that have the same sign, add their absolute values and keep the same sign.


Example 2

Add.a. 27 + 12 b. –16 + (– 22)

Solutiona. 27 + 12 = 39

b. –16 + (–22) = –38


Adding Numbers with Different Signs

To add two numbers that have different signs, subtract the smaller absolute value from the greater absolute value and keep the sign of the number with the greater absolute value.


Example 3

Add.a. 35 + (–17) b. –29 + 7

Solutiona. 35 + (–17) = 18

b. –29 + 7 = –22


Objective 2

Add rational numbers.


Adding Fractions with the Same Denominator

To add fractions with the same denominator, add the numerators and keep the same denominator; then simplify.


Example 4

Add.a. b.

Solutiona.

2 4

9 9

2 4

9 9

2 3 2

3 3 3

4 5

12 12

4 5b.

12 12

3 3 3

3 2 2 4

Replace 6 and 9 with their prime

factorizations, divide out the common factor, 3, then multiply the remaining factors.

Simplify to lowest terms by dividing out the common factor, 3.

2 4 6

9 9

4 5 9

12 12


Adding Fractions

To add fractions with different denominators:1. Write each fraction as an equivalent fraction

with the LCD.2. Add the numerators and keep the LCD.3. Simplify.

Solution


Example 5a1 1

Add: 3 4

1 1

3 4 Write equivalent fractions

with 12 in the denominator.

Add numerators and keep the common denominator.

Because the addends have the same sign, we add and keep the same sign.


Example 5b5 3

Add: 6 4

Write equivalent fractions with 12 in the denominator.

Add numerators and keep the common denominator.

Because the addends have different signs, we subtract and keep the sign of the number with the greater absolute value.

Solution


7 15 9(4)

8 15 30(4)

Example 5c7 9

Add: 8 30

7 9

8 30 Write equivalent fractions

with 120 in the denominator.

105 36

120 120 Add numerators and keep

the common denominator.

105 36

120

Reduce to lowest terms.

69

120

3 23

2 2 2 3 5

23

40

Bank account :


Example 6


Objective 4

Subtract rational numbers.


Parts of a subtraction statement:

8 – 5 = 3

Minuend

Subtrahend

Difference


Rewriting Subtraction

To write a subtraction statement as an equivalent addition statement, change the operation symbol from a minus sign to a plus sign, and change the subtrahend to its additive inverse.


Example 9a

Subtracta. –17 – (–5)

SolutionWrite the subtraction as an equivalent addition.

–17 – (–5)

= –17 + 5 = –12

Change the operation from minus to plus.

Change the subtrahend to its additive inverse.

Solution


Example 9b3 1

Subtract: 8 4

3 1

8 4

1

4

3

8

Write equivalent fractions with the common denominator, 8.

3 1(2)

8 4(2)

3 2

8 8

1

4

3

8

5

8


Multiplying and Dividing Real Numbers; Properties of Real Numbers1.41.4

1. Multiply integers.2. Multiply more than two numbers.3. Multiply rational numbers.4. Find the multiplicative inverse of a number.5. Divide rational numbers.


Objective 1

Multiply integers.


In a multiplication statement, factors are multiplied to equal a product.

ProductFactors

2 3 = 6


Multiplying Two Numbers with Different Signs

When multiplying two numbers that have different signs, the product is negative.


Example 2

Multiply.

a. 7(–4) b. (–15)3

Solutiona. 7(–4) =

b. (–15)3 =

Warning: Make sure you see the difference between 7(–4), which indicates multiplication, and 7 – 4, which indicates subtraction.

–28

–45


Multiplying Two Numbers with the Same Sign

When multiplying two numbers that have the same sign, the product is positive.


Example 3

Multiply.

a. –5(–9) b. (–6)(–8)

Solutiona. –5(–9) =

b. (–6)(–8) =

45

48


Objective 2

Multiply more than two numbers.


Multiplying with Negative Factors

The product of an even number of negative factors is positive, whereas the product of an odd number of negative factors is negative.


Objective 3

Multiply rational numbers.


Example 5a

Multiply

Solution

3 4 .

5 9

3 4 3 2 2

5 9 5 3 3

4

15

Divide out the common factor, 3.

Because we are multiplying two numbers that have different signs, the product is negative.


Objective 5

Divide rational numbers.


Dividend

8 2 = 4

Divisor

Quotient

Parts of a division statement:


Dividing Signed Numbers

When dividing two numbers that have the same sign, the quotient is positive.When dividing two numbers that have different signs, the quotient is negative.


Example 8

Divide.a. b.

Solutiona. b.

56 ( 8)

56 ( 8) 7

72 6

72 6 12


Division Involving 0

0 0 when 0.n n

0 is undefined when 0.n n

0 0 is indeterminate.


Example 9

Divide

Solution

3 4.

10 5

3 4 3 5

10 5 10 4 Write an equivalent multiplication.

3 5

5 2 2 2

Divide out the common factor, 5.

3

8

Because we are dividing two numbers that have different signs, the result is negative.


Exponents, Roots, and Order of Operations1.51.5

1. Evaluate numbers in exponential form.2. Evaluate square roots.3. Use the order-of-operations agreement to simplify numerical expressions.4. Find the mean of a set of data.


Objective 1

Evaluate numbers in exponential form.


Evaluating an Exponential Form

To evaluate an exponential form raised to a natural number exponent, write the base as a factor the number of times indicated by the exponent; then multiply.


Example 1a

Evaluate. (–9)2

SolutionThe exponent 2 indicates we have two factors of –9. Because we multiply two negative numbers, the result is positive.

(–9)2 = (–9)(–9) = 81


Example 1b

Evaluate.

SolutionThe exponent 3 means we must multiply the base by itself three times.

33

5

33

5

3 3 3

5 5 5

27

125


Evaluating Exponential Forms with Negative Bases

If the base of an exponential form is a negative number and the exponent is even, then the product is positive.

If the base is a negative number and the exponent is odd, then the product is negative.


Example 2

Evaluate.a. b. c. d.

4( 3) 43 3( 2) 32


Objective 2

Evaluate square roots.


Roots are inverses of exponents. More specifically, a square root is the inverse of a square, so a square root of a given number is a number that, when squared, equals the given number.

Square RootsEvery positive number has two square roots, a

positive root and a negative root. Negative numbers have no real-number square

roots.


Example 3

Find all square roots of the given number.

Solutiona. 49Answer 7

b. 81Answer No real-number square roots exist.


The symbol, called the radical, is used to indicate finding only the positive (or principal) square root of a given number. The given number or expression inside the radical is called the radicand.

,

25 5

Radicand

RadicalPrincipal Square Root


Example 4

Evaluate the square root.a. b. c. d.

Solution

169 250.6464

81


Objective 3

Use the order-of-operations agreement to simplify numerical expressions.


Order-of- Operations Agreement

Perform operations in the following order:1. Within grouping symbols: parentheses ( ),

brackets [ ], braces { }, above/below fraction bars, absolute value | |, and radicals .

2. Exponents/Roots from left to right, in order as they occur.

3. Multiplication/Division from left to right, in order as they occur.

4. Addition/Subtraction from left to right, in order as they occur.


Example 5a

Simplify.

Solution

26 15 ( 5) 2

26 15 ( 5) 2

26 ( 3) 2

26 ( 6)

32

Divide 15 ÷ (5) = –3

Multiply (–3) 2 = –6

Add –26 + (–6) = –32


Example 5c

Simplify.

Solution

23 5 6 2 1 49

Calculate within the innermost parenthesis.

Evaluate the exponential form, brackets, and square root.

Multiply 5(3).

Add 9 + 15.

Subtract 24 – 7.


Example 7a

Simplify.

Solution

38( 5) 2

4(8) 8

Evaluate the exponential form in the numerator and multiply in the denominator.

Multiply in the numerator and subtract in the denominator.

Subtract in the numerator.

Divide.


Example 7b

Simplify.

Solution

3

9(4) 12

4 (8)( 8)

Because the denominator or divisor is 0, the answer is undefined.


Translating Word Phrases to Expressions1.61.6

1. Translate word phrases to expressions.


Objective 1

Translating word phrases to Expressions

Look at the pg


The key words sum, difference, product, and quotient indicate the answer for their respective operations.

sum of x and 3

x + 3

difference of x and 3

product of x and 3 quotient of x and 3

x – 3

x 3 x 3


Example 1

Translate to an algebraic expression.a. five more than two times a numberTranslation: 5 + 2n or 2n + 5

b. seven less than the cube of a numberTranslation: n3 – 7

c. the sum of h raised to the fourth power and twelveTranslation: h4 + 12


Example 2

Translate to an algebraic expression.a. seven times the sum of a and bTranslation: 7(a + b)

b. the product of a and b divided by the sum of w2 and 4

Translation: ab (w2 + 4) or 2 4

ab

w


Evaluating and Rewriting Expressions1.71.7

1. Evaluate an expression.2. Determine all values that cause an expression to be undefined.3. Rewrite an expression using the distributive property.4. Rewrite an expression by combining like terms.


Objective 1

Evaluate an expression.


Evaluating an Algebraic Expression

To evaluate an algebraic expression:1. Replace the variables with their corresponding

given values.2. Calculate the numerical expression using the order

of operations.


Example 1a

Evaluate 3w – 4(a – 6) when w = 5 and a = 7.

Solution3w – 4(a 6)

3(5) – 4(7 – 6)= 3(5) – 4(1)= 15 – 4= 11

Replace w with 5 and a with 7.

Simplify inside the parentheses first.

Multiply.

Subtract.


Objective 2

Determine all values that cause an expression to be undefined.


Examples:


The Distributive Property of Multiplication over Addition

a(b + c) = ab + ac

This property gives us an alternative to the order of operations.

2(5 + 6) = 2(11) 2(5 + 6) = 25 + 26

= 22 = 10 + 12

= 22