### 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
πth 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.