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}})

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