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Law of Negative Indices

When the indices of pronumerals are negative, we get a negative indices. An expression with a negative index is the reciprocal of the expression with positive index. For example, x-2  =  1/x^2

Earlier we saw the index law of division – xm ÷ xn, where m > n, and obtain the expression xm-n.

What happens when n > m?

When n > m, m – n < 0, or negative. So

24 ÷ 26  =  24-6  =  2-2  =  1/{2^2}  (using index law of division)

Or {2^4}/{2^6}  =  {2~*~2~*~2~*~2}/{2~*~2~*~2~*~2~*~2~*~2}

= 1/{2~*~2}

1/{2^2}

Hence a negative index can be summarised as –

x-m  =  1/{x^m}  where x <> 0, or

x-m is the reciprocal of xm, since x-m x xm 

=  x-m+m  =  x0  =  1

 

Examples:

  1. 2-3  =  1/{2^3}  =  1/8

  2. (2a^3)^-1  =  1/{2a^3}

  3. (2/3)^-2  =  1/{(2/3)^2}

    =  (3/2)^2  =  9/4

  4. {6x^2}/{3x^4}  =  6/3 x {x^2}/{x^4}

    =  2x-2  =  2/{x^2}

 

Tips for using scientific calculator

We use exactly the same steps for the power of index, but with the +/- button before entering the index number. So to enter 2-3 in a scientific calculator, 

 

Here are some more examples of negative indices:

  1. m-3 x m2  =  m-1  =  1/m

  2. (x^2)^-1  =  x-2  =  1/{x^2}

  3. (2x^2)^-2  =   1/(2x^2)^2  =  1/{4x^4}

  4. {10a^3b^2}/{5a^5b}  =  10/5a3-5b2-1

    =  2a-2b  =  2b/a^2

  5. {4x}/y  =  4xy-1