If $\omega$ is an imaginary cube root of unity, then $(3+5\omega+3\omega^{2})^{2}+(3+3\omega+5\omega^{2})^{2}$ is equal to :

Q: If $\omega$ is an imaginary cube root of unity, then $(3+5\omega+3\omega^{2})^{2}+(3+3\omega+5\omega^{2})^{2}$ is equal to :

To solve the problem, we need to evaluate the expression $(3+5\omega+3\omega^{2})^{2}+(3+3\omega+5\omega^{2})^{2}$ given that $\omega$ is an imaginary cube root of unity. The properties of cube roots of unity are crucial here: 1. The cube roots of unity are $1$, $\omega$, and $\omega^2$, where $\omega \neq 1$ and $\omega^3 = 1$. 2. They satisfy the relation $\omega^2 + \omega + 1 = 0$. Let's denote $a = 3 + 5\omega + 3\omega^2$ and $b = 3 + 3\omega + 5\omega^2$. We need to find $a^2 + b^2$. First, simplify $a$ and $b$ using the relation $\omega^2 + \omega + 1 = 0$: - For $a = 3 + 5\omega + 3\omega^2$: \[ a = 3 + 5\omega + 3\omega^2 = 3 + 5\omega + 3(-1 - \omega) = 3 + 5\omega - 3 - 3\omega = 2\omega \] - For $b = 3 + 3\omega + 5\omega^2$: \[ b = 3 + 3\omega + 5\omega^2 = 3 + 3\omega + 5(-1 - \omega) = 3 + 3\omega - 5 - 5\omega = -2 - 2\omega \] Now, calculate $a^2 + b^2$: - $a^2 = (2\omega)^2 = 4\omega^2$ - $b^2 = (-2 - 2\omega)^2 = 4 + 8\omega + 4\omega^2$ Substitute $\omega^2 = -1 - \omega$ into these expressions: - $a^2 = 4(-1 - \omega) = -4 - 4\omega$ - $b^2 = 4 + 8\omega + 4(-1 - \omega) = 4 + 8\omega - 4 - 4\omega = 4\omega$ Now, add $a^2$ and $b^2$: \[ a^2 + b^2 = (-4 - 4\omega) + 4\omega = -4 \] Thus, the expression $(3+5\omega+3\omega^{2})^{2}+(3+3\omega+5\omega^{2})^{2}$ simplifies to $-4$, which corresponds to Option 1. **Explanation for Incorrect Options:** - **Option 2 (-2):** This option is incorrect because the calculations show that the sum of the squares results in $-4$, not $-2$. - **Option 3 (-1):** This option is incorrect because the derived result from the calculations is $-4$, not $-1$. - **Option 4 (0):** This option is incorrect because the expression evaluates to $-4$, not zero. Therefore, the correct answer is indeed Option 1, $-4$.

Question

Accepted Answer

To solve the problem, we need to evaluate the expression $(3+5\omega+3\omega^{2})^{2}+(3+3\omega+5\omega^{2})^{2}$ given that $\omega$ is an imaginary cube root of unity. The properties of cube roots of unity are crucial here:

1. The cube roots of unity are $1$, $\omega$, and $\omega^2$, where $\omega \neq 1$ and $\omega^3 = 1$.
2. They satisfy the relation $\omega^2 + \omega + 1 = 0$.

Let's denote $a = 3 + 5\omega + 3\omega^2$ and $b = 3 + 3\omega + 5\omega^2$. We need to find $a^2 + b^2$.

First, simplify $a$ and $b$ using the relation $\omega^2 + \omega + 1 = 0$:

- For $a = 3 + 5\omega + 3\omega^2$:
  $$
  a = 3 + 5\omega + 3\omega^2 = 3 + 5\omega + 3(-1 - \omega) = 3 + 5\omega - 3 - 3\omega = 2\omega
  $$

- For $b = 3 + 3\omega + 5\omega^2$:
  $$
  b = 3 + 3\omega + 5\omega^2 = 3 + 3\omega + 5(-1 - \omega) = 3 + 3\omega - 5 - 5\omega = -2 - 2\omega
  $$

Now, calculate $a^2 + b^2$:

- $a^2 = (2\omega)^2 = 4\omega^2$
- $b^2 = (-2 - 2\omega)^2 = 4 + 8\omega + 4\omega^2$

Substitute $\omega^2 = -1 - \omega$ into these expressions:

- $a^2 = 4(-1 - \omega) = -4 - 4\omega$
- $b^2 = 4 + 8\omega + 4(-1 - \omega) = 4 + 8\omega - 4 - 4\omega = 4\omega$

Now, add $a^2$ and $b^2$:

$$
a^2 + b^2 = (-4 - 4\omega) + 4\omega = -4
$$

Thus, the expression $(3+5\omega+3\omega^{2})^{2}+(3+3\omega+5\omega^{2})^{2}$ simplifies to $-4$, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (-2):** This option is incorrect because the calculations show that the sum of the squares results in $-4$, not $-2$.
- **Option 3 (-1):** This option is incorrect because the derived result from the calculations is $-4$, not $-1$.
- **Option 4 (0):** This option is incorrect because the expression evaluates to $-4$, not zero.

Therefore, the correct answer is indeed Option 1, $-4$.