__Objective__

At the end of this lesson, students should be able to multiply and divide directed numbers.

The rules for multiplying and dividing two directed number are as follows;

__Example 1__

__Solution__

__Example __

Simplify:

a) (–2)(–3)(–4) b) (–1)(–2)(–1)(–3)(–4)(–2)(–1)

__Solution __

__Exponents__

__Example 1__

**Simplify: **a) (–3)^{2 }b) –3^{2}

__ __

__Solution__

- a) (–3)
^{2}= (–3)(–3) = +**9****(– x – = +)** - b) –3
^{2}= –(3)(3) = (–1)(9) =**–9****(– x + = –)**

Note the difference between the above examples.

In the example b) the square “to the power 2” was *only* on the 3; it was *not* on the minus sign. Parentheses or brackets makes all the difference in the two examples.

__Example 2 __

**Simplify**

**a) (–5)**^{3}b) (–2)^{4}

- a) (–5)
^{3}= (–5)(–5)(–5)

= (25)(–5)

=**–125** - b) (–2)
^{4}= (–2)(–2)(–2)(–2)

= + 4 x + 4

= (4)(4)

= **16**

__Example 3 __

a = –2 and b = –6

Work out the following

a) a^{2 }b)a^{2 }+ b^{2} c) b^{3 }– a^{2} d) (a – b)^{2}

__Solution __

a) a^{2 }= –2 × –2 = **4**

b) a^{2 }+ b^{2} = (–2 × – 2)+( –6 × –6) = 4 + 36 = **40**

c) b^{2 }– a^{2 }= (–6 × –6) – (–2 × –2) = 36 – 4 = **32**

d) (a – b)^{2 }= (–2 – –6)^{2 }= (–2 + 6) = (4)^{2 }= **16**

__B0DMAS__

__Example 1 __

Simplify the following.

__Solution __

In this problem you can see all the different operations. When there is more than one operation involved in a problem, we have to follow the order of operations.

- 1. Brackets
- Division or multiplication from left to right
- Addition or subtraction from left to right

(a) 23 – (5– –7)

(First, carry out the operations inside the brackets, – – = +)

= 23 – (**5** + **7**)

= 23 – (12) (next, remove the brackets )

= 11

(b) 5 × – 4 + –8 (Remember the order, × before +)

= –20 + –8 (+ – = –)

= – 20 – 8

= –28

(c) –25 ÷ [ 5 × –2 – (–3 ×5) ]

(In this problem, ( )brackets are inside the [ ] brackets, so first you have to solve the inner brackets before and the outer brackets.)

= – 25 ÷ [5 × –2 – (–15)] (–3 ×5 = –15 )

= – 25 ÷ [–10 – (–15) ] (5 × –2 = –10)

= –25 ÷ [–10 + 15] (– – = + )

= –25 ÷ [5]

= – 5 (– ÷ + = – )

d) –8 + (6 × –3 – 2 ) ÷ –4 – –8 ( 6 x –3 –2 = –18 –2)

= –8 +(–18 –2 )÷ –4 + 8 –18 –2 =–20

= –8 + –20 ÷ 4 –20 ÷ 4=–5

= –8 – 5

= –13

__Unit 7 lesson 4: Exercise 1__

Try the following

1) 146 x –2

2) –100 x 3

3) –72 x –1

4) 5 x –18

5) 45 ×–23

6) 20 x 30

7) –35 x –5

8) 40 x –7

9) –7 x –7

10) 14 x –10

__ ____Unit 7 lesson 4: Exercise 2__

Evaluate the following

__Unit 7 Lesson 4: Exercise 3__

1) What number do you multiple by –3 to get the following?

a) 6 b) –90 c) 45 d) 81 e) 21

2) What will you get if you divide -48 by the following?

a) –2 b) –8 c) 12 d) 24 e) +4

3) What number do you divide by –36 to get the following?

a) –9 b) 4 c) 12 d) 6 e) –3

4) Write down six different multiplication problems that will give –12 as the answer.

5) Write down six different division problems that will give –4 as the answer.

6) Put the answers to the calculations in order from lowest to

7) Evaluate the following

8) Evaluate the following

9)Evaluate the following

a) 3 + 9 × –6 2 b) 13 x 2 – (6 +7) c) (–2–2) × [6 + (–6)]

10) x = –2, y = –3 and z = – Work out the following:

a) x^{2 }b) y^{2 }+ x^{2} c ) z^{2 }– x^{2 }d) (x – y)^{2}