**ALGEBRA I** **LESSON
#15 **

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**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:**

Thus, **the Multiplicative
Property of 0 states:**

a times 0 = 0 and 0 times a = 0

**And another property with multiplication.****
**

*The Multiplicative Property
of -1
*

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.
Thus the answer is -9xy**

**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 + (-4y^{2 })] 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)]**