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œ”ΒΉΒΉ in its simplest form, where πœ” is a primitive cube root of unity.

02:30

Video Transcript

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

The notation in this question is the Greek letter πœ”. It is not a lowercase w. And πœ” 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œ” is equal to the cube root of one.

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

In this question, we are asked to simplify πœ” 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‘Ž to the power of π‘₯ multiplied by π‘Ž to the power of 𝑦 is equal to π‘Ž to the power of π‘₯ plus 𝑦. If we are multiplying two powers of the same base, we can simply add the two powers. We can therefore rewrite πœ” to the 11th power as πœ” to the ninth power multiplied by πœ” squared, as nine plus two is equal to 11.

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

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