For the equation Z^5=-7776i, we deduce that Z=+6.
To work with the repeating decimals, note that since there are two digits repeating, you will be using a 1 followed by two zeros, which is 100. This factor will be important for subsequent calculations.
Let’s denote the repeating decimal as x.
Thus, you can write
x = 0.212121...
Now, if you multiply this equation by 100, you can write it on top of the previous equation.
This results in:
100x = 21.212121...
x = 0.212121...
Next, set up a subtraction by taking the second equation from the first:
100x = 21.212121...
x = 0.212121...
- ----------------------------
99x = 21
Solving for x involves dividing both sides by 3:
33x = 7
Then, divide both sides by 33:
x = 7/33
Thus, we conclude that the repeating decimal 0.212121... equates to the fraction 7/33.
Answer: the fraction is 7/33
