Write \(\frac{1}{81}\) as a power of \(3\).

Writing a number as a power of \(3\) means expressing it in the form \(3^{k}\). Deal with the \(81\) first, then with the fact that the number is a fraction.

Building up the powers of \(3\) by repeated multiplication: \(3^1=3\), \(3^2=9\), \(3^3=27\), \(3^4=81\). So \(81=3^4\) [M1].

That makes the number \(\frac{1}{3^4}\). A reciprocal of a power is written with a negative index, using \(\frac{1}{a^{n}} = a^{-n}\). Therefore \(\frac{1}{81}=3^{-4}\) [A1].

The negative sign records that the power is on the bottom of the fraction, not that the value is negative. Since \(\frac{1}{81}\) is a small positive number, an answer such as \(-3^4\) would be wrong in sign as well as in form.