Sure! Let's solve the two parts of the question step by step.
Part (a): Write \(\frac{1}{81}\) as a power of a positive integer.
To begin, observe that 81 can be expressed as a power of a positive integer. Specifically, we know:
[tex]\[ 81 = 3^4 \][/tex]
Therefore, we can rewrite \(\frac{1}{81}\) as:
[tex]\[ \frac{1}{81} = \frac{1}{3^4} \][/tex]
Recall that the reciprocal of a power can be written as a negative exponent:
[tex]\[ \frac{1}{3^4} = 3^{-4} \][/tex]
So, \(\frac{1}{81}\) as a power of a positive integer is:
[tex]\[ \boxed{3^{-4}} \][/tex]
Part (b): How many different ways can you write the answer of part (a)?
In part (a), we found that \(\frac{1}{81}\) can be written as \(3^{-4}\) by expressing 81 as \(3^4\). Since our objective is to write \(\frac{1}{81}\) as a power of a positive integer with the simplest form of the exponent, \(3^{-4}\) is the unique representation.
Considering the condition that we are using positive integers as the base and we have appropriately handled the negative exponent, there is only one correct representation. Therefore, the number of different ways to write the answer of part (a) is:
[tex]\[ \boxed{1} \][/tex]
