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 – x^{m} ÷ x^{n}, where m > n, and obtain the expression x^{mn}.
What happens when n > m?
When n > m, m – n < 0, or negative. So
2^{4} ÷ 2^{6} = 2^{46} = 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 x^{m}, since x^{m} x x^{m}
= x^{m+m} = x^{0} = 1
Examples:

2^{3} = =

=

=
= =

= 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:

m^{3} x m^{2} = m^{1} =

= x^{2} =

= =

= a^{35}b^{21}
= 2a^{2}b =

= 4xy^{1}