Negative and fractional indices help you write powers in a different way.
A negative index means a reciprocal, and a fractional index links powers to roots.
Common mistake: A negative index does not make the answer negative. \(3^{-2}\) means \(\dfrac{1}{3^2} = \dfrac{1}{9}\), not \(-9\).
Negative indices
A negative index means “put it underneath” as a reciprocal.
In general, \(a^{-n} = \dfrac{1}{a^n}\).
Worked example:
Evaluate \(8^{-\tfrac{1}{3}}\).
Step 1
A fractional index of \(\tfrac{1}{3}\) means cube root.
\[8^{\tfrac{1}{3}} = 2\]
Step 2
The negative index means use the reciprocal.
\[8^{-\tfrac{1}{3}} = \dfrac{1}{2}\]
Answer:
\[8^{-\tfrac{1}{3}} = \dfrac{1}{2}\]
Fractional indices
A fractional index means a root and a power.
In general, \(a^{\tfrac{1}{n}} = \sqrt[n]{a}\) and \(a^{\tfrac{m}{n}} = \left(\sqrt[n]{a}\right)^m\).
Worked example:
Evaluate \(81^{-\tfrac{3}{4}}\).
Step 1
Find the fourth root first.
\[81^{\tfrac{1}{4}} = 3\]
Step 2
Then cube the result.
\[81^{\tfrac{3}{4}} = 3^3 = 27\]
Step 3
The negative index means use the reciprocal.
\[81^{-\tfrac{3}{4}} = \dfrac{1}{27}\]
Answer:
\[81^{-\tfrac{3}{4}} = \dfrac{1}{27}\]