Indices and Its Various Laws

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.

1. Conversion of numbers to standard form
2. 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

5²

5^3/2 x 5^1/2  = 5^{3/2 + ½}

5²               5²

= 5^4/2 => 5² => 5^2-2 = 5⁰ => 1

5²        5²

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. (2³)² = 2³ x 2³

= 2 x 2 x 2 x 2 x 2 x 2 = 2⁶

(2³)² = 2^3×2 = 2⁶

v. Product power law:- (ab)^x = a^xb^x

E.g. (3pq)² = 3² x p² x q²

= 9p²q²

vi. Negative Index

e.g.      2⁶  . 2⁹ = 2⁶ = 2 x 2 x 2 x 2 x 2 x 2

2⁹    2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

= ¹/₂₃

Also 2⁶  2⁹ = 2^-3

2^-3 = ¹/2³

• Roof Power e.g.

9¹/₂ = 9 = 3

• Fraction Index

e.g. (i)  4^5/2 = ()⁵ = 2⁵ = 32

(ii) 8^4/3 = ()⁴ = 2⁴ = 16

e.g. 3.5682

Integer    decimal fraction