1-6 Multiplying and Dividing Real Numbers Hubarth Algebra.

1-6 Multiplying and Dividing Real Numbers

HubarthAlgebra

Properties

Identity Property of MultiplicationFor every real number n, 1 n = n and n 1 = n

Multiplication Property of ZeroFor every real number n, n 0 = 0 and 0 n = 0

Multiplication Property of -1For every real number n, -1 n = -n and -n -1 = n

Make a Conjecture

22 2 12 0 22(2(

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a. b.

3(2(1(0(-1(-2(-3(

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What can we now determine from the patterns?

Positive times a positive = positivePositive times a negative = negativeNegative times a positive = negativeNegative times a negative = positive

Rule

Multiplying Number With the Same SignsThe product of two positive numbers or two negative numbers is positive.

Examples 5 2 = 10 -5(-2) = 10

Multiplying Numbers With Different SignsThe product of a positive number and a negative number, or a negative and a positive number,is negative.

Examples 3(-6) = -18 -3 6 = -18

Simplify each expression.

a. –3(–11) –3(–11) = 33 The product of two negative numbers is positive.

b. –6( )34

The product of a positive number and a negative number is negative.

–6( ) = –34

184

Ex 1 Multiplying Numbers

= reduce –

Evaluate 5rs for r = –18 and s = –5.

5rs = 5(–18)(–5) Substitute –18 for r and –5 for s.

= –90(–5) 5(–18) results in a negative number, –90.

= 450 –90(–5) results in a positive number, 450.

Ex 2 Evaluating Expressions

Exponents tell us how many times a number is used as a factor.Example = = 625 = = -32

RuleWhen you have a negative factor, the exponent tells you how many negative there are.If you have an odd number of negative the answer will be negative.If you have a even number of negatives your answer will be positive.Example = -27 = 81

*note* the () means the negative is part of the factor where as no () means the negative is separate.

= -( = -9 = (-3 -3) = 9

Use the order of operations to simplify each expression.

a. –24

= –16 Simplify.

= 81 Simplify.

b. (–3)4

Write as repeated multiplication.–(2 • 2 • 2 • 2)=

Write as repeated multiplication.(–3)(–3)(–3)(–3)=

Ex 3 Simplify Exponential Expressions

Rule

Dividing Numbers With The Same SignThe quotient of two positive numbers or two negative numbers is positive.

Example = 2

Divide Numbers With Different SignsThe quotient of a positive and a negative number, or a negative and positive number,is negative.

Example = -2

Simplify each expression.

a. 70 ÷ (–5)

b. –54 ÷ (–9)

The quotient of a positive number and a negative number is negative.= –14

The quotient of a negative number and a negative number is positive.

= 6

Ex 4 Dividing Numbers

Evaluate – – 4z2 for x = 4, y = –2, and z = –4.

= – 4(16) Simplify the power.–4–2

= 2 – 64 Divide and multiply.

= –62 Subtract.

– – 4z2 = – 4(–4)2 Substitute 4 for x, –2 for y, and –4 for z.xy

–4–2

Ex 5 Evaluating Expressions

Property

Inverse Property of MultiplicationFor every nonzero real number a, there is a multiplicative inverse such thata = 1

Examples 5() = 1 -5( - ) = 1

The multiplicative inverse, or reciprocal, of a nonzero rational number is .Zero does not have a reciprocal. Division by zero is undefined.

Evaluate for p = and r = – .

= –2 Simplify.

pr

32

34

= p ÷ r Rewrite the equation.pr

= ÷ Substitute for p and – for r.34

(– )32

32

34

= Multiply by – , the reciprocal of – .43

(– )32

43

34

Ex 6 Division Using the Reciprocal

Practice

1. Simplify each expression.

a. 4(-6) b. -10(-5) c. - ()

2. Evaluate each expression for c= -8 and d= -7

a. –(cd) b. (-2)(-3)(cd)

3. Simplify each expression.

a. - b. c. -42 7 d.

4. Evaluate - for x = 4, y = -2 and z= -4

5. Evaluate for x = and y = -

-24 50 -

-56 336

-64 16 -613

-62

-2