i^0 = 1
i^1 = i
i^2 = -1
i^3 = -i
this pattern repeats every 4 powers.
i^4 = i^0 = 1
i^5 = i^1 = i
i^6 = i^2 = -1
i^7 = i^3 = -i
i^8 = i^0 = 1
i^9 = i^1 = i
i^10 = i^2 = -1
i^11 = i^3 = -i
looking at the pattern, some smart person figured out that if you divide the exponent by 4, then you use the remainder to find the value of i.
for example:
to find the value of i^4, divide 4 by 4 to get a remainder of 0 and so the value of i^4 is the same as the value of i^0 which is equal to 1.
to find the value of i^5, divide 5 by 4 to get a remainder of 1 and so the value of i^5 is the same as the value of i^1 which is equal to i.
to find the value of i^10, divide 10 by 4 to get a remainder of 2 and so the value of i^10 is the same as the value of i^2 which is equal to -1.
etc.
use of this formula will help you solve the higher exponent problems.
your first problem is to find the value of i^19.
take 19 and divide it by 4 and you get a remainder of 3, so the value of i^19 = the value of i^3 which is equal to -i.
now to problem 2:
(-1)^(-21/2)
this can be solved as follows:
(-1)^(-21/2) is the same as:
((-1)^(1/2))^-21).
since (-1)^(1/2) is equal to sqrt(-1), this is the same as:
sqrt(-1)^-21.
since sqrt(-1) is equal to i, this is the same as:
i^-21.
since i^-21 is equal to 1/i^21, this is the same as:
1/i^21.
since i^21 is equal to i^1 which is equal to i, this is the same as:
1/i.
you can rationalize the denominator by multiplying the numerator and the denominator by i/i to get 1/i * i/i = i / i^2 which is equal to i / -1 which is equal to -i.
that should be your solution.
for problem 1, the solution is -i
for problem 2, the solution is also -i.
here's an online calculator where you can check your work.
http://www.mathsisfun.com/numbers/complex-number-calculator.html
here's a good tutorial on complex numbers.
http://www.purplemath.com/modules/complex.htm
here's the calculations i did in writing.
