2-3: Multiplying/Dividing Rational NumbersIdentity Property of Multiplication
For every real number n, 1 ● n = n and n ● 1 = nIN ENGLISH: Multiplying any number by 1 does not
affect the numberExamples:
1 ● (-5) = -5 -5 ● 1 = -5
Multiplication Property of ZeroFor every real number n, 0 ● n = 0 and n ● 0 = 0IN ENGLISH: Anything multiplied by 0 is 0Examples:
0 ● 35 = 0 35 ● 0 = 0
2-3: Multiplying/Dividing Rational NumbersMultiplication Property of -1
For every real number n, -1 ● n = -n and -1 ● -n = nIN ENGLISH: Multiplying any number by -1 flips the
signExamples:
-1 ● (-5) = 5 -1 ● 5 = -5
2-3: Multiplying/Dividing Rational NumbersThis leads us to two rules
Multiplying numbers with the same sign = positive + ● + - ● -
+ +Multiplying numbers with different signs =
negative + ● - - ● +
- -
2-3: Multiplying/Dividing Rational NumbersExample 1: Multiplying numbers
Simplify -9(-4) = 36 Negative ● Negative = Positive 5(-2/3) = -10/3 Positive ● Negative = Negative
YOUR TURN4(-6)-10(-5)-4.9(-8)-2/3(3/4)
-2450
39.2
-½
2-3: Multiplying/Dividing Rational NumbersExample 2: Evaluating Expressions
Evaluate -2xy for x = -20 and y = -3-2xy = -2(-20)(-3) Substitute using ( )
= 40(-3) Multiply left right = -120
YOUR TURNEvaluate for c = -8 and d = -7-(cd)(-2)(-3)(cd)c(-d)
-56336
-56
2-3: Multiplying/Dividing Rational NumbersExample 3: Simplifying Exponential Expressions
PARENTHESIS MATTER!!!-34 Means -1 ● 34
-(3)(3)(3)(3) = -81(-3)4
(-3)(-3)(-3)(-3) = 81YOUR TURN
-43
(-2)4
(-0.3)2
-(¾)2
-6416
0.09
-9/16
2-3: Multiplying/Dividing Rational NumbersThe rules for signs when dividing are the same
as the rules for multiplicationDividing numbers with the same sign = positive
+ ● + - ● - + +
Dividing numbers with different signs = negative + ● - - ● +
- -
2-3: Multiplying/Dividing Rational NumbersExample 4: Dividing numbers
Simplify 12 ÷ (-4) = -3 Positive ÷ Negative = Negative -12 ÷ (-4) = 3 Negative ÷ Negative = Positive
YOUR TURN-42 ÷ 7-8 ÷ (-2)8 ÷ (-8)-39 ÷ (-3)
-64
-1
13
2-3: Multiplying/Dividing Rational NumbersExample 5: Evaluating Expressions
Evaluate –x/-4 + 2y ÷ z for x = -20, y = 6 and z = -1 -x/-4 + 2y ÷ z -(-20)/-4 + 2(6) ÷ (-1) use ( ) 20/-4 + 2(6) ÷ (-1) -(-20) = 20 -5 + (-12) Multiply/Divide -17 Add
YOUR TURNEvaluate for x = -8, y = -5 and z = -13x ÷ (2z) + y ÷ 10
3z2 – 4y ÷ x
-4.5
-1/5
29.5
2
2
z x
y
2-3: Multiplying/Dividing Rational NumbersInverse Property of Multiplication
For every nonzero real number a, there is a multiplicative inverse 1/a such that a(1/a) = 1
IN ENGLISH: Multiplying any number by it’s reciprocal equals 1
Examples: 5(1/5) = 1 -5(-1/5) = 1
Why we need to know this Dividing using fractions isn’t possible, so instead, we
multiply by the reciprocal.
2-3: Multiplying/Dividing Rational NumbersExample 6: Division Using the Reciprocal
Evaluate x/y for x = -3/4 and y = -5/2
x/y = x ÷ y Rewrite for viewing ease
= -3/4 ÷ -5/2 Substitute
= -3/4(-2/5) Rewrite as multiplication
= 3/10
YOUR TURNEvaluate x/y for x = 8 and y = -4/5
-10