a and b are roots of x²+3x+3=0 then (a+1)¹⁰⁰ +(b+2)¹⁰⁴=?

To find the value of \((a+1)^{100} + (b+2)^{104}\), we first need to determine the roots \(a\) and \(b\) of the quadratic equation \(x^2 + 3x + 3 = 0\).

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we have:

\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{-3 \pm \sqrt{9 - 12}}{2} = \frac{-3 \pm \sqrt{-3}}{2} = \frac{-3 \pm i\sqrt{3}}{2}
\]

Thus, the roots are:
\[
a = \frac{-3 + i\sqrt{3}}{2}, \quad b = \frac{-3 - i\sqrt{3}}{2}
\]

Next, we calculate \(a+1\) and \(b+2\):
\[
a + 1 = \frac{-3 + i\sqrt{3}}{2} + 1 = \frac{-1 + i\sqrt{3}}{2}
\]
\[
b + 2 = \frac{-3 - i\sqrt{3}}{2} + 2 = \frac{1 - i\sqrt{3}}{2}
\]

Now, we need to find:
\[
(a+1)^{100} + (b+2)^{104} = \left(\frac{-1 + i\sqrt{3}}{2}\right)^{100} + \left(\frac{1 - i\sqrt{3}}{2}\right)^{104}
\]

Noticing that both terms can be expressed in polar form, we calculate their magnitudes and angles:

1. The magnitude of \(a + 1\):
   \[
   |a + 1| = \sqrt{\left(\frac{-1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1
   \]

2. The angle of \(a + 1\):
   \[
   \theta_a = \tan^{-1}\left(\frac{\sqrt{3}/2}{-1/2}\right) = \tan^{-1}(-\sqrt{3}) = \frac{2\pi}{3}
   \]

Thus:
\[
a + 1 = e^{i\frac{2\pi}{3}}
\]

Similarly for \(b + 2\):
\[
|b + 2| = 1, \quad \theta_b = \tan^{-1}\left(\frac{-\sqrt{3}/2}{1/2}\right) = -\frac{\pi}{3}
\]
Thus:
\[
b + 2 = e^{-i\frac{\pi}{3}}
\]

Now we compute:
\[
(a+1)^{100} = e^{i\frac{200\pi}{3}} \quad \text{and} \quad (b+2)^{104} = e^{-i\frac{104\pi}{3}}
\]

Reducing the angles:
\[
e^{i\frac{200\pi}{3}} = e^{i\frac{2\pi}{3}} \quad \text{and} \quad e^{-i\frac{104\pi}{3}} = e^{-i\frac{4\pi}{3}}
\]

Finally, we find:
\[
e^{i\frac{2\pi}{3}} + e^{-i\frac{4\pi}{3}} = e^{i\frac{2\pi}{3}} + e^{i(-\frac{4\pi}{3})} = 2\cos\left(\frac{2\pi}{3}\right) = 2\left(-\frac{1}{2}\right) = -1
\]

Therefore, the answer is:
\[
\boxed{-1}
\]

Would you like more details on any part of this solution? Here are some related questions to consider:

1. What are the implications of complex roots in polynomial equations?
2. How can you use De Moivre's Theorem in similar problems?
3. What other methods can be used to solve quadratic equations?
4. How do you convert between rectangular and polar forms?
5. What is the geometric interpretation of complex roots?

Explore the solution to (a+1)¹⁰⁰ + (b+2)¹⁰⁴, where a and b are roots of the quadratic equation x² + 3x + 3 = 0. This problem involves complex numbers, polar coordinates, and the application of the quadratic formula. Ideal for high school students.

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