Find the square root of the complex number
18i

Answer

To find the square root of the complex number \( 18i \), we can follow these steps:

### Step 1: Assume the Square Root
Let us assume that the square root of \( 18i \) can be expressed in the form:
\[ \sqrt{18i} = a + bi \]
where \( a \) and \( b \) are real numbers.

### Step 2: Square Both Sides
Now, we square both sides:
\[ 18i = (a + bi)^2 \]
Expanding the right-hand side gives:
\[ 18i = a^2 + 2abi + (bi)^2 \]
Since \( i^2 = -1 \), we can rewrite this as:
\[ 18i = a^2 - b^2 + 2abi \]

### Step 3: Equate Real and Imaginary Parts
Now, we equate the real and imaginary parts from both sides:
- Real part: \( a^2 - b^2 = 0 \)
- Imaginary part: \( 2ab = 18 \)

### Step 4: Solve for One Variable
From the equation \( a^2 - b^2 = 0 \), we can express \( b^2 \) in terms of \( a^2 \):
\[ b^2 = a^2 \]
This implies that \( b = \pm a \).

### Step 5: Substitute into the Imaginary Part
Substituting \( b = \pm a \) into the imaginary part equation \( 2ab = 18 \):
- If \( b = a \):
\[ 2a^2 = 18 \implies a^2 = 9 \implies a = 3 \text{ or } a = -3 \]
Then, \( b = 3 \) or \( b = -3 \).

- If \( b = -a \):
\[ 2a(-a) = 18 \implies -2a^2 = 18 \implies a^2 = -9 \]
This case does not yield real solutions since \( a^2 \) cannot be negative.

### Step 6: Find Corresponding Values of \( b \)
From the valid cases:
1. If \( a = 3 \), then \( b = 3 \).
2. If \( a = -3 \), then \( b = -3 \).

### Step 7: Write the Final Solutions
Thus, we have two possible values for the square root:
\[ \sqrt{18i} = 3 + 3i \quad \text{or} \quad \sqrt{18i} = -3 - 3i \]
We can also express this as:
\[ \sqrt{18i} = \pm 3(1 + i) \]

### Final Answer:
\[ \sqrt{18i} = \pm 3(1 + i) \]

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