Find the number of non-zero integral solutions of the equation `|1-i|^x=2^xdot`

Explanation

To solve the equation \( |1 - i|^x = 2^x \), we will follow these steps:

### Step 1: Calculate the modulus of \( 1 - i \)

The modulus of a complex number \( z = a + bi \) is given by:

\[ |z| = \sqrt{a^2 + b^2} \]

For \( 1 - i \):

\[ |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 2: Rewrite the equation using the modulus

Now we can rewrite the original equation using the modulus we found:

\[ |1 - i|^x = (\sqrt{2})^x \]

Thus, the equation becomes:

\[ (\sqrt{2})^x = 2^x \]

### Step 3: Simplify the equation

We know that \( \sqrt{2} = 2^{1/2} \). Therefore, we can rewrite \( (\sqrt{2})^x \):

\[ (2^{1/2})^x = 2^x \]

This simplifies to:

\[ 2^{x/2} = 2^x \]

### Step 4: Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal to each other:

\[ \frac{x}{2} = x \]

### Step 5: Solve for \( x \)

To solve for \( x \), we can rearrange the equation:

\[ \frac{x}{2} - x = 0 \implies -\frac{x}{2} = 0 \implies x = 0 \]

### Step 6: Determine non-zero integral solutions

The problem asks for non-zero integral solutions. Since we found that \( x = 0 \) is the only solution, there are no non-zero integral solutions.

### Final Answer

The number of non-zero integral solutions of the equation \( |1 - i|^x = 2^x \) is **0**.

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