For the following exercises, use a calculator to help answer the questions. Evaluate (1-i)^k for

For the following exercises, use a calculator to help answer...

Key Concepts

De Moivre's Theorem

De Moivre's Theorem provides a straightforward method for raising complex numbers in polar form to an integer power by stating that [r(cos ? + i sin ?)]^n = r^n (cos(n?) + i sin(n?)). This theorem simplifies the process of exponentiating complex numbers and reveals the periodic nature of the complex exponential function.

Polar Form Representation

The polar form of a complex number expresses the number in terms of its magnitude and angle, typically as r(cos ? + i sin ?) or re^(i?). This representation is particularly useful for multiplication, division, and exponentiation of complex numbers because it directly relates to the geometric interpretation of scaling and rotation in the complex plane.

Complex Numbers

Complex numbers extend the real numbers by incorporating an imaginary unit, denoted by i, which satisfies i² = -1. This concept allows for the representation of numbers in the form a + bi, where a is the real part and b is the imaginary part, enabling operations that include both rotation and scaling in the complex plane.

Transcript

00:01 This is a problem number 43 in which it is being told to evaluate an expression 1 minus i raised to the power k for k equal to 2 6 10 and 14 when k equal to 2 6 10 and 14.

00:32 Let us start for when k equal to 2 it becomes 1 minus i whole square that is a minus b whole square which is a square plus b square minus 2 a b which is one i square is minus 1 minus 2 i this will get cancelled out so it will be minus 2 i so for k equal to 2 1 minus i square will be minus 2 i okay b is that is number second is 4k equal to 6 for k equal to 6 this means we need to find 1 plus i raised to the power 6 1 plus not 1 plus 1 minus i square raise the power 3 and value of 1 minus i square is minus 2 i 3 so it is minus 2 q this is minus 8 i square into i this will be minus 8 into i this will be minus 8 into 2 minus 1 into i so 8 i so when k equal to 6 the value becomes 8 i now we have third third part when k equal to 10 so 1 minus i raise to the power 10 it will become 1 minus i square raised to the power 5 that is minus 2 i raised to the power 5 this will be minus 2 raise to the power 5 i raise to the power 5 so this is minus 32 i raised to the power 4 into i so i raise to power 4 is 1 because i square equal to minus 1 so i raise to the power 4 equal to 1 so this will be equal to minus 32 i so we have when k equal to 10 this is minus 32 i now, fourth is, we need to evaluate when k equal to 14.

03:11 When k equal to 14, 1 minus i.

03:15 Raised to the power 14 will be 1 minus i.

03:20 Raised to the power 2 into 7, sorry.

03:27 So this will be minus 2 i raised to the power 7, which will be minus 2 raised to the power 7, which will be minus 2 raised to the power 7 into i raise to the power 7...

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