Find the square root of `a^(2)-1+2ai`

Answer

To find the square root of the expression \( a^2 - 1 + 2ai \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ a^2 - 1 + 2ai \]
We can rewrite \(-1\) as \(i^2\) since \(i^2 = -1\):
\[ a^2 + i^2 + 2ai \]

### Step 2: Recognize the perfect square
Now, we can recognize that the expression resembles the expansion of a perfect square. The expression \(a^2 + 2ai + i^2\) can be factored using the formula for a square:
\[ (a + i)^2 \]

### Step 3: Take the square root
Now that we have expressed the original expression as a perfect square, we can take the square root:
\[ \sqrt{(a + i)^2} \]
The square root and the square cancel each other out, giving us:
\[ a + i \]

### Final Answer
Thus, the square root of \( a^2 - 1 + 2ai \) is:
\[ \boxed{a + i} \]