Indices is a measure of the price performance of a group of stocks on an exchange. Trading indices allows you to access the entire economy or industry at once, and only need to open a position. Indices are a useful way to express large numbers more simply. They also provide us with many useful properties for manipulating them using the so-called Law of Indices.

- Conversion of numbers to standard form
- Law of indices, indicial equation

**Example:**__ 1 __ = 27^{(1 – x)}

** **81^{(x – 2)}

** **

** **

** **

**Solution:**

__ 1 __ = 27^{(1 – x)}

81^{(x – 2)}

__ 1 __ = 3^{3(1 – x)}

3^{4(x – 2)}

3^{-4(x – 2)} = 3^{3(1 – x)}

-4(x – 2) = 3(1 – x)

-4x + 8 = 3 – 3x

4x – 3x = 8 – 3

x = 5

** Example 2:**

** **625^{3/8} x 5^{1/2 } 25

__625 ^{3/8} x 5^{1/2}__

5^{2}

__5 ^{3/2} x 5^{1/2} __ =

__5__

^{3/2 + ½}5^{2} 5^{2}

= __5 ^{4/2}__ =>

__5__=> 5

^{2}^{2-2}= 5

^{0}=> 1

5^{2} 5^{2}

i. Multiplication rule

X^{a} x X^{b} = X^{a+b}

ii. Division rule

X^{a} / X^{b} = X^{a-b}

^{ }

iii. Zero Index

X^{O} = 1, X 0

iv. Power law:- (a^{x})^{y} = a^{xy}

E.g. (2^{3})^{2} = 2^{3} x 2^{3}

= 2 x 2 x 2 x 2 x 2 x 2 = 2^{6}

(2^{3})^{2} = 2^{3×2} = 2^{6}

v. Product power law:- (ab)^{x} = a^{x}b^{x}

E.g. (3pq)^{2} = 3^{2} x p^{2} x q^{2}

= 9p^{2}q^{2}

vi. Negative Index

e.g. 2^{6 } . 2^{9} = __2 ^{6} __=

__2 x 2 x 2 x 2 x 2 x 2__

2^{9} 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

= ^{1}/_{23}

Also 2^{6} 2^{9} = 2^{-3}

2^{-3} = ^{1}/2^{3}

- Roof Power e.g.

9^{1}/_{2} = 9 = 3

- Fraction Index

e.g. (i) 4^{5/2} = ()^{5} = 2^{5} = 32

(ii) 8^{4/3} = ()^{4} = 2^{4} = 16

e.g. 3.5682

Integer decimal fraction