If `x+iy=(1+i)(1+2i)(1+3i)`, then `x^(2)+y^(2)` equals:

Options

A. `-1`
B. 0
C. 1
D. 100

Answer

To solve the problem \( x + iy = (1+i)(1+2i)(1+3i) \) and find \( x^2 + y^2 \), we can follow these steps:

### Step 1: Calculate the modulus of the product
Using the property of modulus, we know that:
\[ |z_1 z_2| = |z_1| \cdot |z_2| \]
Thus, we can write:
\[ |x + iy| = |(1+i)(1+2i)(1+3i)| = |1+i| \cdot |1+2i| \cdot |1+3i| \]

### Step 2: Calculate the modulus of each complex number
Now we will calculate the modulus of each term:
1. For \( |1+i| \):
\[ |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]
2. For \( |1+2i| \):
\[ |1+2i| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]
3. For \( |1+3i| \):
\[ |1+3i| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

### Step 3: Combine the moduli
Now we can combine these results:
\[ |x + iy| = |1+i| \cdot |1+2i| \cdot |1+3i| = \sqrt{2} \cdot \sqrt{5} \cdot \sqrt{10} \]

### Step 4: Simplify the expression
We can simplify this product:
\[ |x + iy| = \sqrt{2 \cdot 5 \cdot 10} = \sqrt{100} = 10 \]

### Step 5: Relate modulus to \( x^2 + y^2 \)
Since \( |x + iy| = \sqrt{x^2 + y^2} \), we can square both sides to find \( x^2 + y^2 \):
\[ x^2 + y^2 = |x + iy|^2 = 10^2 = 100 \]

### Final Answer
Thus, the value of \( x^2 + y^2 \) is:
\[ \boxed{100} \]