__INDICES__

Hi students,
are you with me? Alright, let’s start with **Indices**.

__Definition:__

**Indices** is the plural form of **index** and **index** means power. Therefore, **indices** means powers.
The **index notation** is used as shorthand for the further related multiplication of the number by itself e.g.

2³ = 2 x 2 x
2

2^{4}
= 2 x 2 x 2 x 2 and of course 2¹ = 2

In general α¹
= α

α² = α
x α

α³ = α
x α x α

α^{п}
= α x α x α x α …… α, n times.

__Laws of Indices__

**Law
1: **α^{m} x α^{п}

= α^{п+1}

This
law, theoretically, means to multiply powers of a number, add the indices e.g.

5³ x 5^{4}

= 5^{3+4}

= 5^{7}//.

# Laws 2: α^{m}
÷ α^{n}

= α^{m-n}

This law, theoretically,
states; for the division of the powers of the same number, the indices are subtracted e.g.

7^{5} ÷ 7^{3}

= 7^{5
- 3 }

= 7^{2}//

**Law
3:** (α^{m})^{n}

= α^{m
x n}

Theoretically,
indices are multiplied when the power of a number is itself raised to another power e.g.

(5 ³)²

= 5 ³ ^{x} ²

= 5^{6}

*THE ZERO POWER
*

**Law
4**: n° = 1

Theoretically,
any number raised to the power of Zero is equals to one. E.g.

3^{4}
÷ 3^{4}

= 3^{4 - 4}

= 3°

= 1

*NEGATIVE INDICES
*

**Law
5**: α^{-n}

= __1__

α^{n}

The result of dividing 4^{3} ÷ 4^{5} using the laws of indices is;

4^{3}
÷ 4^{5}

=
4^{3-5}

= 4^{-2}

If we use the
notation of the division line, we can see what negative index means at once.

__4³__

4^{5}

= __4__^{1}
x 4^{1} x 4^{1}

^{1}4 x ^{1}4 x ^{1}4 x 4 x 4

=^{
}__1__

4^{2}

= 4^{-2}//.

The negative
sign is not an indication that the number is negative, but simply that a certain power of number has been divided by a larger
power.

*FRACTIONAL INDICES
*

**Law
6:** α^{1/n} = ^{n} α (for all values of an excluding zero)
e.g

i. 9^{½ }ii. 4^{– ½ }

= √9
= __1__

4^{½
}

= 3 =
__1__ __1 __

√4 = 2
or 2^{-1}//.

**Law
7:** α^{m/n} = (n√α)^{m} or ^{n}√α^{m}
e.g

i. 27²/³ ii.
8²/³

= (3√27)^{2}
= (3√8)^{2}

= 3^{2}
= 2^{2}

= 9//. = 4

__Note:__ In law 6, the square root of a certain number means, the digit that can multiply itself two times to
obtain the number in the radical sign. Likewise in law 7, the cube root of a number is the digit that can multiply itself
three times to obtain the number in the radical sign. Same thing applies to both 4^{th} root, 5^{th} root
etc.

Now that you’ve
comprehended the laws of indices, let’s proceed to solve some problems based on these laws. I want the student to understand
that he cannot solve any indices problem without comprehending the laws. However, my advice to my students is this; go through
the laws over and over until you can memorise them without this paper. The laws are stated below again more clearly for your
comprehension.

### Laws of Indices

- α
^{m} x α^{n} = α^{m+n}
- α
^{m} ÷ α^{n} = α^{m-n}
- (α
^{m})^{n} = α^{m x n}
- α
^{o} = 1
- α
^{–n} = 1

α^{n}

- α
^{1/n} = √α
- α
^{m/n} = (n√α)^{n} or n√α^{n}//.

*EXAMPLES *

1. Simplify
α^{2} b^{3} x α^{4} b^{5}

# Solution

α^{2} b^{3} x α^{4} b^{5 }

= α^{2}
x α^{4} x b^{3} x b^{5} (collecting like terms)

= α^{2+4}
x b^{3+5} (applying law 1)

= α^{6}
b^{8}//.

2. Simplify
__χ__^{5} y^{2}

χ^{2}
y

**Solution
**

__χ__^{5 }y^{2}

χ^{2 }y

= χ^{5}
÷ χ^{2} x y^{2} ÷ y^{1} (collecting like terms)

= χ^{5-2
}x y^{2-1 }(applying law 2)

= χ^{3}
y

or __χ__^{5}
y^{2 }

χ^{2}
y

= χ^{3}
y

3. Simplify
__α__^{4} b^{3} + α^{5} b^{6}

α^{2}
b

**Solution**

__α__^{4} b^{3 }+ α^{5} b^{6}

α^{2} b

= __α__^{4}
b^{3} _{+} __α__^{5} b^{6}

α^{2}
b α^{2} b

α^{4}
÷ α^{2} x b^{3} ÷ b^{1} + α^{5} ÷ α^{2} x b^{6} (collecting
like terms)

α^{4-2}
x b^{3-1} + α^{5-2} x b^{6-1} (applying law 2)

= α^{2}
b^{2} + α^{3} b^{5}//.

Or __α__^{4}
b^{3} + __α__^{5} b^{6}

α^{2}b

= __α__^{4}
b^{3} + __α__^{5} b^{6}

α^{2}
b α^{2} b

=
α^{2} b^{2} + α^{3} b^{5}//.

4.
Simplify (χ^{-2})^{3}

**Solution**

(χ^{-2})^{3}

= χ^{-2
x 3}

= χ^{6}

5.
Simplify __4__^{-½} x 16^{¾}

4^{½}

**Solution** __4__^{–½} x 16^{¾}

4^{½}

=
__1__

__4__^{½} x 16^{¾ }(Applying law 5)

4^{½
}

=
__1 __x (4√16)^{3}

__√4
__(Applying law 6 and 7)

√4

=
__1/2 x 2__^{3} (Removing Radical sign)

2

=
__1/2 x 8__

2

=
__8/2__

2

=
__4__

2

=
2//.