**Definition of Integer**

An integer is any positive or negative whole number

Table of Contents

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**Example:**

Simplify the following

(+8) + (+3) (ii) (+9) – (+4)

Solution

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

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**Evaluation**

Simplify the following

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

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**Indices**

The plural of index is indices

10 x 10 x 10= 10^{3} in index form, where 3 is the index or power of 10. P^{5}=p x pxpxpxp. 5 is the power or index of p in the expression P^{5}.

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**Laws of Indices**

- Multiplication law:

a^{x} x a^{y} = a^{x+y}

E.g. a^{5}xa^{3}=a x a x a x a x a x a x a x a =a^{8}

y^{1 }x y^{4}=y^{ 1+4}

= y^{5}

a^{3 }x a^{5 }= a^{3 + 5 }= a^{8}

4c^{4 }x 3c^{2}

= 4 x 3 x c^{4 }x c^{2 }=12 x c^{4+2}=12c^{6}

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**Class work **

Simplify the following

(a) 10^{3 }x 10^{4 }(b) 3 x 10^{6 }x 4 x 10^{2 }(c) p^{3 }x p (d) 4f^{3 }x 5f^{7 }

^{ }

**Division law**

(1) a^{x }÷ a^{y }= a^{x }÷ a^{y }= a^{x-y}

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**Example**

Simplify the following

- a
^{7}÷a^{3}=a x a x a x a x a x a x a ÷ a x a x a

a^{7-3}=a^{4}

(2) 10^{6}÷10^{3}=10^{6}÷10^{3}=10^{6-3}=10^{3}

(3) 10a^{7}÷2a^{2}=10a^{7}÷2a^{2}=5a^{7-2}=5a^{5}

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**Class work**

Simplify the following

- 10
^{5}÷10^{3}2. 51m^{9}÷3m (3) 8×10^{9}÷4×10^{6 }

**Zero indexes**

a^{x }÷ a^{x }=1

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**By division law** a^{x-x}=a^{0}

a^{0}=1

E.g. 100^{0 }=1

50^{0}=1

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**Negative index**

a^{0 }÷ a^{x }= 1/a^{x}

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

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

**Example**

- Simplify (i) b
^{-2}(ii) 2^{-3}

**Solution**

b^{-2} = 1/ b^{2 } (ii) 2^{-3} = 1/2^{3 }= 1/2x2x2 = 1/8

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**Class work**

(1) 10^{-2} (2) d^{0 }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. [a^{2}]^{4}= x a^{2 }x a^{2 }x a^{2 }= a x a x a x a x a x a x a x a=a^{8}

Therefore. a^{2×4}=a^{8.}

(6) [mn]^{ a}=m ^{a}x n^{a }= m^{a}n^{a}. e.g. [4+2x]^{ 2}=4^{2}+2^{2}xx^{2} =

16+4x^{2}=4[4+1xx^{2}] =4[4+x^{2}].

7 Fractional indexes

a^{m/n} =a^{1/n }x^{m}=^{n}√ am

Example

(a^{1/2})^{2} =a^{2/2}=a^{1}=a

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

√32^{1}

- 32
^{3/5 = 5}√2^{5×3 }= 2^{3 }=2x2x2 =__8__ - 27
^{2/3}=^{3}√27^{2 }= 3^{2 }= 3x3x3 = 9 - 4
^{-3/2 }= √1/4^{3}=__1/2__^{3} - (0001)
^{3}

=1×10^{-3}

=(10^{-3)3=}10^{-3×3=}10^{-9}

__= 1 .__

1000000000

=0.000000001

- (a
^{m})^{p/q}=a^{mp}=√(a)^{p}

e.g. (16^{2})^{3/4}=√ (16^{2})^{3}

= (2^{2})^{3}

(4)^{3=}4x4x4 = 64

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- Equator of power for equal base

A^{x}=A^{y} That is x = y

**WEEKEND ASSIGNMENT **

- Simplify (+13) (+6)

(a)7 (b) -7^{ }_{ }(c) 19 (d) 8

- Simplify (+11) (+6)- (-3)

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

- Simplify 5x
^{3}x 4x^{7 }(a) 20x^{4 }(b) 20x^{10 }(c) 20x^{7}(d) 57x^{10} - Simplify 10a
^{8}÷ 5a^{6}(a) 2a^{2}(b) 50a^{2 }(c) 2a^{14 }(d) 2a^{48} - Simplify r
^{7}÷ r^{7}(a) 0 (b) 1 (c) r^{14 }(d) 2r^{7}

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**THEORY **

**Simplify**

- 5y
^{5}x 3y^{3} - 24×8

6x

**Simplify**(1/2)^{-3}

**See also**