Let `z = ((2)/(i + sqrt(3)))^(200)+((2)/(i-sqrt(3)))^(200),"then" |z + (1.7)^(2)|` = ________________.

Answer

To solve the problem, we need to evaluate the expression \( z = \left( \frac{2}{i + \sqrt{3}} \right)^{200} + \left( \frac{2}{i - \sqrt{3}} \right)^{200} \) and then find \( |z + (1.7)^2| \).

### Step-by-Step Solution:

1. **Rewrite the fractions**:
We start with the expression for \( z \):
\[ z = \left( \frac{2}{i + \sqrt{3}} \right)^{200} + \left( \frac{2}{i - \sqrt{3}} \right)^{200} \]

2. **Multiply numerator and denominator by \( i \)**:
To simplify the denominators, we multiply the numerator and denominator by \( i \):
\[ z = \left( \frac{2i}{i^2 + \sqrt{3}i} \right)^{200} + \left( \frac{2i}{i^2 - \sqrt{3}i} \right)^{200} \]
Here, \( i^2 = -1 \), so we have:
\[ z = \left( \frac{2i}{-1 + \sqrt{3}i} \right)^{200} + \left( \frac{2i}{-1 - \sqrt{3}i} \right)^{200} \]

3. **Simplify the denominators**:
The terms simplify to:
\[ z = \left( \frac{2i}{\sqrt{3}i - 1} \right)^{200} + \left( \frac{2i}{-\sqrt{3}i - 1} \right)^{200} \]

4. **Express in terms of \( \omega \)**:
Let \( \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \) (which is a cube root of unity). Then:
\[ z = \left( \frac{2i}{\omega} \right)^{200} + \left( \frac{2i}{\omega^2} \right)^{200} \]

5. **Calculate powers of \( i \)**:
Since \( i^{200} = (i^4)^{50} = 1^{50} = 1 \), we can simplify:
\[ z = \frac{2^{200} i^{200}}{\omega^{200}} + \frac{2^{200} i^{200}}{\omega^{400}} = \frac{2^{200}}{\omega^{200}} + \frac{2^{200}}{\omega^{400}} \]

6. **Use properties of \( \omega \)**:
We know \( \omega^3 = 1 \), hence \( \omega^{400} = \omega^{1} \) and \( \omega^{200} = \omega^{2} \):
\[ z = 2^{200} (\omega^{-2} + \omega^{-1}) = 2^{200} (\omega + \omega^2) \]

7. **Sum of cube roots of unity**:
Since \( \omega + \omega^2 = -1 \):
\[ z = 2^{200} \cdot (-1) = -2^{200} \]

8. **Calculate \( |z + (1.7)^2| \)**:
Now we need to find \( |z + (1.7)^2| \):
\[ (1.7)^2 = 2.89 \]
Thus:
\[ |z + 2.89| = |-2^{200} + 2.89| \]

9. **Final calculation**:
Since \( 2^{200} \) is a very large number, we can approximate:
\[ | -2^{200} + 2.89 | \approx 2^{200} \]

### Conclusion:
The final answer is:
\[ |z + (1.7)^2| = 2^{200} - 2.89 \approx 2^{200} \]