Question

Problem: Report Error We have $64^{-1} = \frac{1}{64}.$ But $\frac{1}{64}$ can be written as an integer raised to an integer power in other ways, too! How many different ways in total can $\frac{1}{64}$ be written as an integer raised to an integer power, including $64^{-1}$?

Answer 1

Answer:

4 different ways

Step-by-step explanation:

Given $64^{-1} = \frac{1}{64}$, we are told that the expression cam also be written as an integer raised to an integer power in other ways, the other ways are as shown below;

First way:

$(2^6)^{-1} = 2^{-6}\\2^{-6} = \dfrac{1}{2^6} \\Hence \ 64^{-1} = 2^{-6}$

Second way:

$(64)^{-1} = (4^3)^{-1}\\4^{-3} = \dfrac{1}{4^3} \\Hence \ 64^{-1} = 4^{-3}$

third way:

$(64)^{-1} = (8^2)^{-1}\\8^{-2} = \dfrac{1}{8^2} \\Hence \ 64^{-1} = 8^{-2}$

Therefore the expression $64^{-1}$ can also be written as $2^{-6}, 4^{-3} \ and \ 8^{-2}$.

The total number of different ways $\frac{1}{64}$ can be written including $64^{-1}$ is 4