How would you solve an imaginary number with -31 as its exponent?

Answer 1

The following discussion is for complex numbers; this includes (pure) imaginary numbers as a special case. This type of powers (a complex number to the power of a real number) are very simple if you write the complex number in polar coordinates, specifying an absolute value and an angle. Raise the absolute value to the specified power, and multiply the angle by the power.

Example (writing on a piece of paper is clearer; it is difficult to represent some of the symbols here):

(1 + i)6 = [(square root of 2) angle (45 degrees)]

Square root of 2 to the power 6 is 8.

45 degrees x 6 = 270 degrees, which is the same as minus 90 degrees.

The result is, then, 8 at an angle of -90 degrees. Converting this back to rectangular coordinates, this is equal to -8i.

Answer 2

You need the basic property of i, which is i^2 = -1.

If you are asking about i^31, we proceed as follows.

Then i^31 = i*(i^2)^15 = i*(-1)^15 = i*(-1) = -i

If the number is ai, which is pure imaginary, then

(ai)^31 = (a^31)*(i^31) = -(a^31)*i