Let `z = (1)/(2) (sqrt(3) + i),"then" |(z^(101)+i^(103))^(105)|`= ________.

Answer

To solve the problem, we need to evaluate the expression \( |(z^{101} + i^{103})^{105}| \) where \( z = \frac{1}{2}(\sqrt{3} + i) \).

### Step 1: Find the modulus of \( z \)

First, we calculate the modulus of \( z \):
\[ |z| = \left| \frac{1}{2}(\sqrt{3} + i) \right| = \frac{1}{2} |\sqrt{3} + i| \]
Now, we find \( |\sqrt{3} + i| \):
\[ |\sqrt{3} + i| = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]
Thus,
\[ |z| = \frac{1}{2} \cdot 2 = 1 \]

### Step 2: Calculate \( z^{101} \)

Since \( |z| = 1 \), we have:
\[ |z^{101}| = |z|^{101} = 1^{101} = 1 \]

### Step 3: Calculate \( i^{103} \)

Next, we calculate \( i^{103} \). The powers of \( i \) cycle every 4:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

To find \( i^{103} \), we compute \( 103 \mod 4 \):
\[ 103 \div 4 = 25 \quad \text{(remainder 3)} \]
Thus, \( i^{103} = i^3 = -i \).

### Step 4: Combine \( z^{101} \) and \( i^{103} \)

Now we combine the results:
\[ z^{101} + i^{103} = z^{101} - i \]
Since \( |z^{101}| = 1 \) and \( |i| = 1 \), we need to find the modulus of the sum \( z^{101} - i \).

### Step 5: Calculate the modulus of \( z^{101} - i \)

Using the triangle inequality:
\[ |z^{101} - i| \leq |z^{101}| + |i| = 1 + 1 = 2 \]
To find the exact modulus, we note that \( z^{101} \) lies on the unit circle, and \( -i \) also lies on the unit circle. The distance between two points on the unit circle can be calculated using the formula:
\[ |z^{101} - i| = \sqrt{|z^{101}|^2 + |i|^2 - 2|z^{101}||i|\cos(\theta)} \]
where \( \theta \) is the angle between \( z^{101} \) and \( -i \). Since both have modulus 1, we can simplify:
\[ |z^{101} - i| = \sqrt{1 + 1 - 2\cos(\theta)} = \sqrt{2(1 - \cos(\theta))} \]
This is maximized when \( \theta = \pi \) (i.e., when they are opposite on the unit circle), giving:
\[ |z^{101} - i| = \sqrt{2(1 - (-1))} = \sqrt{4} = 2 \]

### Step 6: Calculate the modulus of the entire expression

Now we can find the modulus of the entire expression:
\[ |(z^{101} + i^{103})^{105}| = |(z^{101} - i)^{105}| = |z^{101} - i|^{105} \]
Since we found \( |z^{101} - i| = 2 \):
\[ |(z^{101} - i)^{105}| = 2^{105} \]

### Final Answer

Thus, the final answer is:
\[ \boxed{2^{105}} \]