# Negative Integral Indices

The Negative Indices
$${a^{ - n}}$$ is a reciprocal of $${a^{n}}$$. ${a^{ - n}} = \frac{1}{{{a^n}}}$
Example
Find the value of the following.
a. $${102^{ - 1}}$$
b. $$- {6^{ - 3}}$$
c. $${\left( {\frac{1}{3}} \right)^{ - 4}}$$
d. $${\left( {\frac{2}{5}} \right)^{ - 2}}$$
e. $${\left( { - \frac{2}{5}} \right)^{ - 4}}$$

a. $${102^{ - 1}} = \frac{1}{{120}}$$
b. $$- {6^{ - 3}} = \frac{1}{{ - {6^3}}} = - \frac{1}{{216}}$$
c. $${\left( {\frac{1}{3}} \right)^{ - 4}} = {\left( 3 \right)^4} = 81$$
d. $${\left( {\frac{2}{5}} \right)^{ - 2}} = {\left( {\frac{5}{2}} \right)^2} = \frac{{25}}{4}$$
e. $${\left( { - \frac{2}{5}} \right)^{ - 4}} = {\left( { - \frac{5}{2}} \right)^4} = \frac{{625}}{{16}}$$