If the complex number `omega=x+iy(AA x, y in R and i^(2)=-1)` satisfy the equation `omega^(3)=8i`, then the maximum vlaue of y is

Options

A. 1
B. `(sqrt3)/(2)`
C. `(1)/(2)`
D. 2

Answer

To solve the problem, we need to find the maximum value of \( y \) for the complex number \( \omega = x + iy \) that satisfies the equation \( \omega^3 = 8i \).

### Step-by-Step Solution:

1. **Express the equation**: We start with the equation given:
\[ \omega^3 = 8i \]
where \( \omega = x + iy \).

2. **Expand \( \omega^3 \)**: Using the binomial theorem, we expand \( (x + iy)^3 \):
\[ (x + iy)^3 = x^3 + 3x^2(iy) + 3x(iy)^2 + (iy)^3 \]
This simplifies to:
\[ = x^3 + 3x^2(iy) + 3x(i^2y^2) + i^3y^3 \]
Knowing that \( i^2 = -1 \) and \( i^3 = -i \), we have:
\[ = x^3 + 3x^2(iy) - 3xy^2 - iy^3 \]
Rearranging gives:
\[ = (x^3 - 3xy^2) + i(3x^2y - y^3) \]

3. **Set equal to \( 8i \)**: Now, we equate the real and imaginary parts to those of \( 8i \):
\[ x^3 - 3xy^2 = 0 \quad \text{(real part)} \]
\[ 3x^2y - y^3 = 8 \quad \text{(imaginary part)} \]

4. **Solve the real part equation**: From the real part equation:
\[ x^3 = 3xy^2 \]
If \( x \neq 0 \), we can divide both sides by \( x \):
\[ x^2 = 3y \quad \Rightarrow \quad x = \sqrt{3y} \quad \text{or} \quad x = -\sqrt{3y} \]

5. **Substitute into the imaginary part**: We substitute \( x^2 = 3y \) into the imaginary part equation:
\[ 3(3y)y - y^3 = 8 \]
This simplifies to:
\[ 9y^2 - y^3 = 8 \]
Rearranging gives:
\[ y^3 - 9y^2 + 8 = 0 \]

6. **Find the roots of the polynomial**: We can use the Rational Root Theorem or synthetic division to find the roots of the polynomial. Testing \( y = 1 \):
\[ 1^3 - 9(1^2) + 8 = 1 - 9 + 8 = 0 \]
So \( y = 1 \) is a root. We can factor the polynomial as:
\[ (y - 1)(y^2 - 8y - 8) = 0 \]

7. **Solve the quadratic**: Now we solve \( y^2 - 8y - 8 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{64 + 32}}{2} = \frac{8 \pm \sqrt{96}}{2} = \frac{8 \pm 4\sqrt{6}}{2} = 4 \pm 2\sqrt{6} \]

8. **Determine the maximum value of \( y \)**: The values of \( y \) are \( 1 \), \( 4 + 2\sqrt{6} \), and \( 4 - 2\sqrt{6} \). The maximum value among these is:
\[ 4 + 2\sqrt{6} \]
However, we need to check if this value is valid in the context of the original equations. Since \( 2\sqrt{6} \approx 4.899 \), we find that \( 4 + 2\sqrt{6} \) is greater than \( 1 \).

9. **Conclusion**: The maximum value of \( y \) that satisfies the equation \( \omega^3 = 8i \) is:
\[ \boxed{1} \]