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Definition of Integer

An integer is any positive or negative whole number

 

Example:

Simplify the following

(+8) + (+3)      (ii) (+9) –  (+4)

Solution

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

 

Evaluation

Simplify the following

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

 

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.

 

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

ax a5 = a3 + 5 = a8

4c4 x 3c2

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

 

Class work

Simplify the following

(a) 103 x 104              (b) 3 x 106 x 4 x 102       (c) p3 x p          (d) 4f3 x 5f7        

 

Division law

(1)  ax ÷ ay = ax ÷ ay = ax-y

 

Example

Simplify the following

  • a7÷a3=a x a x a x a x a x a x a ÷ a x a x a

a7-3=a4

(2) 106÷103=106÷103=106-3=103

(3) 10a7÷2a2=10a7÷2a2=5a7-2=5a5

 

Class work

Simplify the following

  1. 105÷103 2.  51m9÷3m              (3) 8×109÷4×106            

 

Zero indexes

ax ÷ ax =1

 

By division law ax-x=a0

a0=1

E.g. 1000 =1

500=1

 

Negative index

a0 ÷ 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/ b2               (ii) 2-3 = 1/23     = 1/2x2x2 = 1/8

 

Class work

(1) 10-2       (2) d0 x d4 x d-2                (3) a-3÷a-5         (4)  (1/4)-2

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

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

7      Fractional indexes

am/n   =a1/n xm=n√ am

 

Example

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

(√a)2=√a x √a =√a x a=√a2=ae.g321/5=5

√321

  1. 323/5 = 5√25×3 = 23 =2x2x2 = 8
  2. 272/3=3√272 = 32 = 3x3x3 = 9
  3. 4-3/2 = √1/43= 1/23
  4. (0001)3

=1×10-3

=(10-3)3=10-3×3=10-9

=        1           .

1000000000

=0.000000001

 

  1. (am)p/q=amp=√(a)p

e.g. (162)3/4=√ (162)3

= (22)3

 

(4)3=4x4x4 = 64

 

  1. Equator of power for equal base

Ax=Ay That is x = y

 

WEEKEND ASSIGNMENT

  1. Simplify (+13) – (+6)

(a)7  (b) -7   (c) 19    (d) 8

  1. Simplify (+11) – (+6)- (-3)

(a)7   (b)8      (c)9     (d)10

  1. Simplify 5x3 x 4x7 (a) 20x4 (b) 20x10            (c) 20x7           (d) 57x10
  2. Simplify 10a8 ÷ 5a6 (a) 2a2 (b) 50a2 (c) 2a14                (d) 2a48
  3. Simplify r7 ÷ r7 (a) 0 (b) 1     (c) r14    (d) 2r7

 

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

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