Video Transcript
Write 𝜔 to the 11th power in its
simplest form, where 𝜔 is a primitive cube root of unity.
The notation in this question is
the Greek letter 𝜔. It is not a lowercase w. And 𝜔 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 𝜔 is equal to the cube root of one.
It is clear that one value of 𝜔 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 𝜔 is equal to the cube root
of one. We can also express the
relationship as 𝜔 cubed is equal to one.
In this question, we are asked to
simplify 𝜔 to the 11th power. In order to do this, we begin by
recalling one of our rules of exponents or indices. This states that 𝑎 to the power of
𝑥 multiplied by 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 plus
𝑦. If we are multiplying two powers of
the same base, we can simply add the two powers. We can therefore rewrite 𝜔 to the
11th power as 𝜔 to the ninth power multiplied by 𝜔 squared, as nine plus two is
equal to 11.
Next, we can rewrite 𝜔 to the
ninth power as 𝜔 cubed multiplied by 𝜔 cubed multiplied by 𝜔 cubed. This means that 𝜔 to the 11th
power can be rewritten as shown. We know from our definition of the
cube roots of unity that 𝜔 cubed is equal to one, which means that 𝜔 to the 11th
power is equal to 𝜔 squared. Using our knowledge of the cube
roots of unity, the expression, written in its simplest form, is 𝜔 squared.
