ALGEBRA I                                          LESSON #15

If you need to contact your math teacher for additional help, stop by the chatroom or use the Math Response Form.

MULTIPLICATION PROPERTIES & MULTIPLYING REAL NUMBERS

The equation     3 x 0 = 0 and 0 x 3 = 0 illustrate the theorem called the:

Multiplicative Property of Zero:
When one of the factors of a product is 0, the product itself is 0.

Thus, the Multiplicative Property of 0 states:
For each real number a,

a times 0 = 0   and  0 times a = 0

And another property with multiplication.

The Multiplicative Property of   -1
Multiplying any real number by -1 produces the additive inverse of the number.

Thus, the Multiplicative Property of  -1
For each real number a,

a(-1) = -a     and   (-1)a  = -a

A special case of this property occurs when a = -1, you have:  (-1)(-1) = 1 The following examples show how to use the multiplicative property of -1, along with the multiplication facts for positive numbers, and the associative and communicative axioms, to multiply any two real numbers.

Examples:       7 x  3 = 21

(-7)3 = (-1 x 7)3 = -1(7 x  3) = -1(21) = -21

7(-3) - 7[3(-1)] = (7  x  3)(-1) = 21(-1) = -21

(-7)(-3) = (-1  x  7)(-1  x  3) = [-1(-1)](7 x  3) = 1  x  21 = 21

These examples suggest the following theorem:

Property of Opposites in Products

For all real numbers a and b,

(-a)b = -ab,   a(-b) = -ab,      (-a)(-b) = ab

Practice in simplifying products should lead you to discover the following rules for multiplication:

Rules for Multiplication

1.  The absolute value of the product of two or more real numbers is the product of the absolute values of the members:
|ab| = |a| times |b|.

2.  A product of nonzero real numbers of which an even number are negative is a positive number.  A product of nonzero real numbers of which an odd number are negative is a negative number. Examples:  Simplify each expression.

a.  6x + (-7x)                          b.  -19 x  0  x  (-4)                           c.  -3xy + (-6xy)

a.  6x + (-7x) = 6x = (-7)x = [6 + (-7)]x = (-1)x = -x.

b.  -19  x  0 is 0 and 0  x  -4 is 0.  Anything multiplied by 0 is 0.

c.  Since both terms have like variables xy, we can add since there are two negatives.  In addition when you have two negatives, you add and the answer is always negative.

Before you start your work, a few more examples

-84(14) + 16(-14)

---First multiply -84 x 14= 1176 but it will be negative because of the -84, so it will be -1176.
---Now multiply the 16 x -14 which gives you 224, a negative 224 or -224.  Since both the -1176 and
-224 are negative, we will add them together which gives us the answer of 1400, but again because
of the two negatives in addition, the answer will be in the negative,
so the final answer is -1400.

-5[3x + (-2y)]

We will apply the distributive property because the -5 is outside of the brackets.  So we will multiply the -5 with 3x and then -5 with -2y.  Again remember to apply the -5 to everything in the brackets.  So, -5 times 3x is -15x and -5 times -2y is -10y.  Since these two terms -15x and -10y are not like terms, we will leave them as is.  Thus the answer is -15x + -10y.

NOW YOU TRY J

Simplify each expression

1.  (-52 + 49)13

2.  25[13 + (-19)]

3.  53 + 53(-22)

4.  (-63)(-14) + 14(-63)

5.  -3(5a + 2b)

6.  8[ -p +( -4)q]

7.  -5[ -3y + (-4y2 )]    Note this is with an exponent.  If you browser does not support
exponents  you  will see -5[-3y + (-4y2)] We will use a ^ to represent
the exponent.
So the problem should read   -5[-3y + (-4y^2)].

8.   9d + (-2e) + (-6d) + e

9. -2ef + 5ef  + (-5ef) + ef

10-6[4(-3 + 2bc) + ( -2)] + (-4)(-2bc + 1)

11.    -2[ -3 + 5( -3rx + 2)] + 3[11 + ( -5rx)]

12.   5[ -x + 2(3x + y)] + 2[ -y + 3(2y + x)]