__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 + = –)**

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**

**1) a) (–5) ^{3} b) (–2)^{4}**

1) 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**

__BODMAS__

__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.

(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

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