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. \({\left( { - \frac{2}{5}} \right)^{ - 4}}\)

Answer:
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. \({\left( { - \frac{2}{5}} \right)^{ - 4}} = {\left( { - \frac{5}{2}} \right)^4} = \frac{{625}}{{16}}\)