# Question Video: ο»ΏReciprocals of the πth Roots of Unity Mathematics

Let π be an πth root of unity. Which of the following is the correct relationship between π to the power negative one and π? [A] πβ»ΒΉ = βπ [B] πβ»ΒΉ = (π)^* [C] πβ»ΒΉ = π [D] πβ»ΒΉ = β(π)^*

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### Video Transcript

Let π be an πth root of unity. Which of the following is the correct relationship between π to the power negative one and π? Is it (A) π to the power negative one is negative π? (B) π to the power negative one is the complex conjugate of π. (C) π to the power negative one is actually π. Or (D) π to the power negative one is the negative complex conjugate of π. Express π to the power negative one in terms of positive powers of π.

Weβre told that π, which is a complex number, is an πth root of unity. And recall that if complex number π is an πth root of unity, then itβs a solution to the equation π to the power π is equal to one. Thatβs for a positive integer π. And there are two parts to this question. The first is whatβs the correct relationship between π to the power negative one and π, and weβre then asked to express π to the power negative one in terms of positive powers of π. So letβs look first at the first part.

We know that π is an πth root of unity. And recall it from de Moivreβs theorem, we can express these roots in exponential form; that is, π is π to the ππ, where π is two ππ over π and π takes values from zero to π minus one. Since we want to know the relationship between π and π to the power negative one, we can also express π to the power negative one in exponential form. π to the power negative one is equal to π to the power negative ππ, where π is Eulerβs number. And thatβs equal to π to the power π times negative π. And now, if we look at this in trigonometric form, we have π to the power negative one is the cos of negative π plus π times the sin of negative π.

But now we can use the fact that for an angle π, the cos of negative π is the cos of π and the sin of negative π is negative the sin of π so that π to the power negative one is the cos of π minus π sin π in trigonometric form. And this is simply the complex conjugate of π since π is cos π plus π sin π in trigonometric form. So in fact, we have π to the power negative one is the complex conjugate of π; that is, option (B) π to the negative one is equal to the complex conjugate of π.

We can check that none of the other options apply. So, for example, option (A) says that π to the negative one is equal to negative π. But we can see, in fact, from the exponential form that this is not true, so we can discount option (A). Option (C) is that π to the negative one is equal to π. This again is not true because we have π to the negative ππ against π to the ππ, so these are not equal and we can discount option (C). And option (D) says that π to the negative one is the negative complex conjugate of π. But the negative complex conjugate of π is negative cos π minus π sin π, which is negative one times cos π plus π sin π, which is not equal to π to the power negative one. So we can discount option (D).

Now, for the second part of our question, weβre asked to express π to the power negative one in terms of positive powers of π. We could look again at the exponential form, but letβs instead look at what we mean by π to the power negative one in terms of powers or exponents. We know that the laws of exponents for real numbers tell us that π to the power negative one is one over π if π is a real number. And in fact, this applies also to complex numbers. So in fact, π to the power negative one is equal to one over π. And, of course, π is never zero since zero is never a root of unity.

But letβs look again at our equation π to the power π is equal to one. If we divide through by π, we have π to the power π over π is one over π. But thatβs equal to π to the negative one. And now if we were to write this out in full, we have π to the power π over π is π times itself π times over π. And we can simple this by canceling the π in the denominator with one on the top. So now, in our numerator, we have π times itself π minus one times and one in the denominator, which is actually π to the power π minus one so that π to the power negative one is π to the power π minus one.

Remember that π is a positive integer so that for any π greater than one, we have π to the negative one in terms of a positive power of π. And if π is equal to one, we have π minus one equal to zero. And in this case, in our equation, π to the power π is equal to one; thatβs the πth root of unity. That means π is the first root of unity, which is one. And so if π is an πth root of unity, π to the power negative one is the complex conjugate of π, thatβs option (B), and π to the power negative one in terms of positive powers of π is π to the power π minus one.