Evaluate 𝜔 minus 𝜔 squared to the power of 14 where 𝜔 is a complex cube root of unity.
We begin by recalling that there are three cubic roots of unity. One of those three roots is one, which is known as the trivial root of unity. And the other two roots are called the complex or nontrivial roots of unity. If we consider the multiplication cycle for the cubic roots of unity, we see that 𝜔 cubed is equal to one. It also follows that 𝜔 is equal to 𝜔 to the fourth power, 𝜔 to the seventh power, and so on. And equivalent expressions to 𝜔 squared and 𝜔 cubed are as shown.
In this question, we need to simplify and then find a value of 𝜔 minus 𝜔 squared to the power of 14. Using our laws of exponents or indices, we can rewrite this as 𝜔 minus 𝜔 squared all squared all raised to the power of seven. Let’s consider the expression inside the square brackets. We can square 𝜔 minus 𝜔 squared using the FOIL method. This is equal to 𝜔 squared minus 𝜔 cubed minus 𝜔 cubed plus 𝜔 to the fourth power, which simplifies to 𝜔 squared minus two 𝜔 cubed plus 𝜔 to the fourth power.
We need to raise this to the seventh power, but before doing so, we can simplify some of the terms. We can replace 𝜔 to the fourth power with 𝜔, and we know that 𝜔 cubed is equal to one. The expression inside the bracket becomes 𝜔 squared plus 𝜔 minus two multiplied by one so that the overall expression is equal to 𝜔 squared plus 𝜔 minus two all raised to the power of seven.
Next, we recall one of the properties of the cube roots of unity, that is, 𝜔 squared plus 𝜔 plus one equals zero. Subtracting one from both sides, we see that 𝜔 squared plus 𝜔 is equal to negative one. This means that our expression simplifies to negative one minus two all raised to the power of seven. This is equal to negative three to the power of seven, which is equal to negative 2,187. The expression 𝜔 minus 𝜔 squared all raised to the power of 14, where 𝜔 is a complex cube root of unity, evaluates to negative 2,187.