Simiplify : `sqrt(-64).(3 + sqrt(-361))`

Answer

To simplify the expression \( \sqrt{-64} \cdot (3 + \sqrt{-361}) \), we will follow these steps:

### Step 1: Identify the square roots of negative numbers
First, we recognize that both \( \sqrt{-64} \) and \( \sqrt{-361} \) can be expressed in terms of the imaginary unit \( i \), where \( i = \sqrt{-1} \).

### Step 2: Simplify \( \sqrt{-64} \)
We can write:
\[ \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i \]
because \( \sqrt{64} = 8 \).

### Step 3: Simplify \( \sqrt{-361} \)
Similarly, we simplify \( \sqrt{-361} \):
\[ \sqrt{-361} = \sqrt{361} \cdot \sqrt{-1} = 19i \]
because \( \sqrt{361} = 19 \).

### Step 4: Substitute back into the expression
Now we substitute these values back into the original expression:
\[ \sqrt{-64} \cdot (3 + \sqrt{-361}) = 8i \cdot (3 + 19i) \]

### Step 5: Distribute \( 8i \)
Next, we distribute \( 8i \) across the terms inside the parentheses:
\[ 8i \cdot 3 + 8i \cdot 19i = 24i + 152i^2 \]

### Step 6: Substitute \( i^2 \) with \(-1\)
Since \( i^2 = -1 \), we can replace \( 152i^2 \) with \(-152\):
\[ 24i + 152(-1) = 24i - 152 \]

### Final Result
Thus, the simplified expression is:
\[ 24i - 152 \]