# BASIC OPERATION OF INTEGERS

## Definition of Integer

An integer is any positive or negative whole number

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Example:

Simplify the following

(+8) + (+3)Â Â Â Â Â  (ii) (+9) –Â  (+4)

Solution

(+8) + (+3) = +11Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (ii) (+9) – (+4) = 9-4 = +5 or 5

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## Evaluation

Simplify the following

(+12) Â–(+7)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (ii) 7-(-3)-(-2)

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## Indices

The plural of index is indices

10 x 10 x 10= 103 in index form, where 3 is the index or power of 10. P5=p x pxpxpxp. 5 is the power or index of p in the expression P5.

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## Laws of Indices

1. Multiplication law:

ax x ay = ax+y

E.g. a5xa3=a x a x a x a x a x a x a x a =a8

y1 x y4=y 1+4

= y5

a3Â  x a5 = a3 + 5 = a8

4c4 x 3c2

= 4 x 3 x c4 x c2 =12 x c4+2=12c6

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### Class work

Simplify the following

(a) 103 x 104Â Â Â Â Â Â Â Â Â Â Â Â Â  (b) 3 x 106 x 4 x 102Â Â Â Â Â Â  (c) p3 x pÂ Â Â Â Â Â Â Â Â  (d) 4f3 x 5f7Â Â Â Â Â Â Â Â

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## Division law

(1)Â  ax Ã· ay = ax Ã· ay = ax-y

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Example

Simplify the following

• a7Ã·a3=a x a x a x a x a x a x a Ã· a x a x a

a7-3=a4

(2) 106Ã·103=106Ã·103=106-3=103

(3) 10a7Ã·2a2=10a7Ã·2a2=5a7-2=5a5

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### Class work

Simplify the following

1. 105Ã·103 2.Â  51m9Ã·3mÂ Â  Â Â Â Â Â Â Â Â Â Â  (3) 8×109Ã·4×106Â Â Â Â Â Â Â Â Â Â Â Â

## Zero indexes

ax Ã· ax =1

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By division law ax-x=a0

a0=1

E.g. 1000 =1

500=1

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## Negative index

a0 Ã· ax = 1/ax

But by division law, a0-x=a-x

Therefore, a-x=1/ax

Example

1. Simplify (i) b-2 (ii) 2-3

Solution

b-2 = 1/ b2Â Â  Â Â Â Â Â Â Â Â Â Â Â  (ii) 2-3 = 1/23Â Â Â Â  = 1/2x2x2 = 1/8

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### Class work

(1) 10-2 Â Â Â Â Â  (2) d0 x d4 x d-2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (3) a-3Ã·a-5Â  Â Â Â Â Â Â  (4)Â  (1/4)-2

(5)Â Â Â Â  [am]n = amxn = amn.

[Power of index]

E.g. [a2]4= x a2 x a2 x a2 = a x a x a x a x a x a x a x a=a8

Therefore. a2×4=a8.

(6)Â Â  [mn] a=m ax na = mana. e.g. [4+2x] 2=42+22xx2 =

16+4x2=4[4+1xx2] =4[4+x2].

7Â Â Â Â Â  Fractional indexes

am/nÂ Â  =a1/n xm=nâˆš am

Example

(a1/2)2 =a2/2=a1=a

(âˆša)2=âˆša x âˆša =âˆša x a=âˆša2=ae.g321/5=5

âˆš321

1. 323/5 = 5âˆš25×3 = 23 =2x2x2 = 8
2. 272/3=3âˆš272 = 32 = 3x3x3 = 9
3. 4-3/2 = âˆš1/43= 1/23
4. (0001)3

=1×10-3

=(10-3)3=10-3×3=10-9

=Â Â Â Â Â Â Â  1Â Â Â Â Â Â Â Â Â Â  .

1000000000

=0.000000001

1. (am)p/q=amp=âˆš(a)p

e.g. (162)3/4=âˆš (162)3

= (22)3

(4)3=4x4x4 = 64

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1. Equator of power for equal base

Ax=Ay That is x = y

## WEEKEND ASSIGNMENT

1. Simplify (+13) Â– (+6)

(a)7Â  (b) -7Â  Â (c) 19Â Â Â  (d) 8

1. Simplify (+11) Â– (+6)- (-3)

(a)7Â Â  (b)8Â Â Â Â Â  (c)9 Â Â Â  (d)10

1. Simplify 5x3 x 4x7 (a) 20x4 (b) 20x10Â Â Â Â Â Â Â Â Â Â Â  (c) 20x7 Â Â Â Â Â Â Â Â Â  (d) 57x10
2. Simplify 10a8 Ã· 5a6 (a) 2a2 (b) 50a2 (c) 2a14Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (d) 2a48
3. Simplify r7 Ã· r7 (a) 0 (b) 1 Â Â Â  (c) r14Â Â Â  (d) 2r7

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## THEORY

1. Simplify
• 5y5 x 3y3
• 24×8

6x

1. Simplify (1/2)-3

See also

WEIGHT

VOLUME OF CYLINDER

AREA OF RIGHT ANGLED TRIANGLE

PROFIT AND LOSS PERCENT

SIMPLE INTERREST