Understanding the Cube Root of Unity

The question asks us to find the value of an expression involving $\omega$, which is a cube root of unity. A cube root of unity is a complex number that, when cubed, equals 1. The three cube roots of unity are $1$, $\omega$, and $\omega^2$, where $\omega = e^{i2\pi/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $\omega^2 = e^{i4\pi/3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$. The fundamental property of the non-real cube roots of unity is that their sum is zero:

$$1 + \omega + \omega^2 = 0$$

This property is key to simplifying expressions involving $\omega$ and $\omega^2$. Another important property is $\omega^3 = 1$. These properties are derived from the roots of unity concept in complex numbers.

Simplifying the Expression

We need to simplify the expression $(3 + \omega + 3\omega^2)^6$. Let's focus on the base of the power first: $3 + \omega + 3\omega^2$. We can use the property $1 + \omega + \omega^2 = 0$ to simplify this expression using basic algebra.

We can rewrite $3 + \omega + 3\omega^2$ as follows:

$$3 + \omega + 3\omega^2 = (1 + 1 + 1) + \omega + (\omega^2 + \omega^2 + \omega^2)$$

This doesn't look very helpful. Let's try grouping terms differently, aiming to use the $1 + \omega + \omega^2 = 0$ property. A common technique is to adjust coefficients to match $1 + \omega + \omega^2$.

$$3 + \omega + 3\omega^2 = (3 + 3\omega + 3\omega^2) - 2\omega$$

Now, we can factor out 3 from the first three terms:

$$3(1 + \omega + \omega^2) - 2\omega$$

Since we know that $1 + \omega + \omega^2 = 0$, we can substitute this into the expression:

$$3(0) - 2\omega = 0 - 2\omega = -2\omega$$

So, the expression inside the parenthesis simplifies to $-2\omega$. This demonstrates how understanding the properties of the cube root of unity is essential for solving such problems in mathematics.

Calculating the Power

Now that we have simplified the base, we need to raise it to the power of 6:

$$ (3 + \omega + 3\omega^2)^6 = (-2\omega)^6 $$

Using the property $(ab)^n = a^n b^n$, we can write:

$$ (-2\omega)^6 = (-2)^6 \cdot (\omega)^6 $$

Let's calculate each part separately:

• Calculate $(-2)^6$: $$ (-2)^6 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2) $$ $$ (-2)^6 = 4 \times 4 \times 4 = 16 \times 4 = 64 $$
• Calculate $(\omega)^6$: We know that $\omega^3 = 1$ because $\omega$ is a cube root of unity. We can write $\omega^6$ as a power of $\omega^3$: $$ (\omega)^6 = (\omega^3)^2 $$ Substitute $\omega^3 = 1$: $$ (\omega^3)^2 = (1)^2 = 1 $$

Now, multiply the results of the two parts:

$$ (-2)^6 \cdot (\omega)^6 = 64 \cdot 1 = 64 $$

Thus, the value of $(3 + \omega + 3\omega^2)^6$ is 64.

Summary of Calculation Steps

Here is a quick summary of the calculation:

1. Identify the expression involving the cube root of unity: $ (3 + \omega + 3\omega^2)^6 $.
2. Use the property $1 + \omega + \omega^2 = 0$ to simplify the base $3 + \omega + 3\omega^2$. $$ 3 + \omega + 3\omega^2 = 3(1 + \omega + \omega^2) - 2\omega = 3(0) - 2\omega = -2\omega $$
3. Substitute the simplified base back into the expression: $ (-2\omega)^6 $.
4. Apply the power to both the coefficient and $\omega$: $ (-2)^6 \cdot (\omega)^6 $.
5. Calculate the numerical power: $ (-2)^6 = 64 $.
6. Calculate the power of $\omega$ using $\omega^3=1$: $ \omega^6 = (\omega^3)^2 = (1)^2 = 1 $.
7. Multiply the results: $ 64 \times 1 = 64 $.

The final value of the expression is 64. This type of problem is common in algebra when dealing with complex numbers and powers of the cube root of unity.