d Recall that i^2=-1. How can you obtain the next powers of i?
A
a 7i
B
b isqrt(2)
C
c -16
D
d - 27i
Practice makes perfect
a We want to calculate the value of the given numeric expression using the imaginary number i. To do so, we will first recall that the imaginary unit i is the number whose square is - 1.
i^2=- 1
In our case, the square root of a negative number is not real, but it can be written using the imaginary unit. This allows us to rewrite the given expression.
b We again want to calculate the value of the given expression. To do so, we can — similar as in Part A — use the fact that sqrt(a)* i = sqrt(- a).
d Similarly, we want to evaluate the value of the given expression containing imaginary number i. To do so, first note that when the base of a power is a product, the expression can be rewritten by putting the exponent on both factors.
To find the power of i, keep in mind that the Commutative and Associative Properties of Multiplication hold true for imaginary numbers. This allows us to rewrite the power as a product.
i^3 = i^2 * i
Now — by the definition of imaginary unit — we know that i^2= - 1.
i^2&= - 1
i^3&= i^2 * i= - 1 * i= - i
Therefore, indeed i^3=- i. Finally, let's use this fact to simplify the given expression.
