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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  = 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  = (using index law of division)

Or = =  Hence a negative index can be summarised as –

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  = = 2. = 3. = = = 4. = x =  2x-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  = 2. =  x-2  = 3. = = 4. = a3-5b2-1

=  2a-2b  = 5. =  4xy-1