BASIC OPERATION OF INTEGERS

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:

a^x x a^y = a^x+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

a³  x a⁵ = a^{3 + 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)  a^x ÷ a^y = a^x ÷ a^y = a^x-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

a^7-3=a⁴

(2) 10⁶÷10³=10⁶÷10³=10^6-3=10³

(3) 10a⁷÷2a²=10a⁷÷2a²=5a^7-2=5a⁵

 

Class work

Simplify the following

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

 

Zero indexes

a^x ÷ a^x =1

 

By division law a^x-x=a⁰

a⁰=1

E.g. 100⁰ =1

50⁰=1

 

Negative index

a⁰ ÷ a^x = 1/a^x

But by division law, a^0-x=a^-x

Therefore, a^-x=1/a^x

 

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)     [a^m]ⁿ = a^mxn = a^mn.

[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. a^2×4=a^8.

(6)   [mn] ^a=m ^ax n^a = m^an^a. e.g. [4+2x] ²=4²+2²xx² =

16+4x²=4[4+1xx²] =4[4+x²].

7      Fractional indexes

a^m/n   =a^1/n x^m=ⁿ√ am

 

Example

(a^1/2)² =a^2/2=a¹=a

(√a)²=√a x √a =√a x a=√a²=ae.g32^1/5=⁵

√32¹

1. 32^{3/5 = 5}√2^5×3 = 2³ =2x2x2 = 8
2. 27^2/3=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

 

8. (a^m)^p/q=a^mp=√(a)^p

e.g. (16²)^3/4=√ (16²)³

= (2²)³

 

(4)³⁼4x4x4 = 64

 

9. Equator of power for equal base

A^x=A^y That is x = y

 

WEEKEND ASSIGNMENT

1. Simplify (+13) – (+6)

(a)7  (b) -7   (c) 19    (d) 8

2. Simplify (+11) – (+6)- (-3)

(a)7   (b)8      (c)9     (d)10

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

 

THEORY

1. Simplify

• 5y⁵ x 3y³
• 24×8

6x

2. Simplify (1/2)^-3

 

See also

WEIGHT

VOLUME OF CYLINDER

AREA OF RIGHT ANGLED TRIANGLE

PROFIT AND LOSS PERCENT

SIMPLE INTERREST