Theory of Indices

The 3 laws of indices for positive integers :

1. a^m x aⁿ = a^m+n
2. a^m ÷ aⁿ = a^m-n     [m>n]
3. (a^m)ⁿ = a^mn

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The law of index for zero (0) :

2⁰ = 2¹ ÷ 2¹     [by law of indices for positive integer]
        2¹ ÷ 2¹ = 1
        \ 2⁰ = 1

In general :

a⁰ = a¹ ÷ a¹     [by law of indices for positive integer]
     = aⁿ ÷ aⁿ = 1
       \ a⁰ = 1

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The law of index for negative integers :

2^-1 = 2⁰ ÷ 2¹     [by law of indices for positive integer]
      = 1 ÷ 2 = ½
        \ 2^-1 = ½

In the same way

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The law of index for fractions :

4^½ = ?
          4^½ x 4^½ = 4¹
          \ (4^½)² = 4      [by law of indices for positive integer]
            \ Ö4 = 4^½

= ?
          x x = 8
          \ ()³ = 8      [by law of indices for positive integer]
            \ =

In general :

\

\

= ?	or	=
()2    [by law of indices for positive integer]		[by law of indices for positive integer]
= 22		=
= 4		= 4

In general :

or

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