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### Lesson 11

Multiplying and dividing real numbers

Properties of real numbers

- 1.multiplication property of -1
- For every real number a,
- a x -1 = -1 x a = -a
- 2. multiplication property of 0
- For every real number a,
- a x 0 = 0
- 3. Inverse property of multiplication
- For every real number a, where a is not equal to 0,
- a x 1/a = 1/a x a = 1

Multiplying signed numbers

- The product of 2 numbers with the same sign is a positive number
- The sign of 2 numbers with different signs is a negative number

Raising a number to a power

- (-3)4 = (-3)(-3)(-3)(-3)= 81
- (-3)3 = (-3)(-3)(-3)= - 27
- -34= - 3 3 3 3= - 81
- -53 = - 5 5 5 = -125
- (-5)3 = (-5)(-5)(-5)= -125

Dividing signed numbers

- 1. The quotient of 2 numbers with the same sign is positive.
- 2. the quotient of 2 numbers with different signs is negative

simplify

- -72/ -24
- 12.1/ -11
- -5/6
- -2/9

Lesson 12using the properties of real numbers to simplify expressions

- The properties of real numbers are used to simplify expressions and write equivalent expressions.

Properties of addition and multiplication

- IDENTITY PROPERTY OF ADDITION
- For every real number a,
- a + 0 = a
- IDENTITY PROPERTY F MULTIPLICATION
- For every real number a,
- a x 1 = a
- COMMUTATIVE PROPERTY OF ADDITION
- For every real number a and b,
- a + b = b + a
- COMMUTATIVE PROPERTY OF MULTIPLICATION
- For every real number a and b,
- a x b = b x a

Properties cont.

- ASSOCIATIVE PROPERTY OF ADDITION
- For every real number a, b, and c,
- (a + b) + c = a + ( b+ c)
- a + b + c = a + b + c
- ASSOCIATIVE PROPERT OF MULTIPLICATION
- For every real number a, b and c,
- (a b) c= a (b c)
- a b c= a b c

Identifying properties

- 1. 1 x 8 = 8
- 2. 6 + 9 = 9 + 6
- 3. (2 x 5) x 4 = 2 x ( 5 x 4)
- 4. (7 + 1) + 3 = (1 + 7) + 3
- 5. 0 + 10 = 10
- 6. 17 x 1 = 17

True or false

- d x 0 = d
- (e r )b= (r e )b
- x (y + z) = (x +y )z
- d + ( e + f) = ( d + e) + f
- How can you be sure???

Lesson 15distributive property

- For all real numbers a, b, and c
- a (b + c) = a b + ac
- And a (b - c) = a b - ac

Distributing positive and negative numbers

- Simplify:
- 6(4+8)= 6(4) + 6(8)
- = 24 + 48 = 72
- 6(2-7) = 6(2) + 6(-7)= 6(2) -6(7)
- =12 + -42 = 12 -42 = -30
- -(9+ 4)= -1(9) + -1 ( 4)
- = -9 + -4 = -13
- -20(12+8)= -20(12) + -20(8)
- = -240 + -160= -400

Simplifying algebraic expressions

- m n( m x + n y+ 2p)= mnmx+mnny+mn2p
- = m2nx + mn2y +2mnp
- (m x +2ny +z) m n= m n m x + mn2ny +m n z
- = m2nx + 2mn2y + m n z
- (y3-zx3)(-yz) = (-yz)y3 +(-yz)(-zx3)
- = -y4z + x3yz2

Check for understanding

- 1. Give an example of an expression that can be simplified using the distributive property.
- 2.How do properties help you to simplify expressions?
- 3. Explain the difference in finding the solution to (-2)2 and -22.

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