Question Video: Evaluating Powers of the Cube Root of Unity | Nagwa Question Video: Evaluating Powers of the Cube Root of Unity | Nagwa

# Question Video: Evaluating Powers of the Cube Root of Unity Mathematics • Third Year of Secondary School

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Write πΒΉΒΉ in its simplest form, where π is a primitive cube root of unity.

02:30

### Video Transcript

Write π to the 11th power in its simplest form, where π is a primitive cube root of unity.

The notation in this question is the Greek letter π. It is not a lowercase w. And π has a specific meaning when we are dealing with complex numbers. It is what we call a cube root of unity and satisfies the equation π is equal to the cube root of one.

It is clear that one value of π is one, since one multiplied by one multiplied by one is equal to one. But, in general, a number has π πth roots, which may be real or complex. This means that there are two further roots, both of which are complex, that satisfy π is equal to the cube root of one. We can also express the relationship as π cubed is equal to one.

In this question, we are asked to simplify π to the 11th power. In order to do this, we begin by recalling one of our rules of exponents or indices. This states that π to the power of π₯ multiplied by π to the power of π¦ is equal to π to the power of π₯ plus π¦. If we are multiplying two powers of the same base, we can simply add the two powers. We can therefore rewrite π to the 11th power as π to the ninth power multiplied by π squared, as nine plus two is equal to 11.

Next, we can rewrite π to the ninth power as π cubed multiplied by π cubed multiplied by π cubed. This means that π to the 11th power can be rewritten as shown. We know from our definition of the cube roots of unity that π cubed is equal to one, which means that π to the 11th power is equal to π squared. Using our knowledge of the cube roots of unity, the expression, written in its simplest form, is π squared.

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