To evaluate the expression \((13 + \sqrt{-225}) - (\sqrt{49} - 2i)\), let's break it down step by step.
Step 1: Simplify \(\sqrt{-225}\)
\[ \sqrt{-225} = \sqrt{225} \cdot \sqrt{-1} = 15i \] So, \(13 + \sqrt{-225} = 13 + 15i\).
Step 2: Simplify \(\sqrt{49}\)
\[ \sqrt{49} = 7 \] Now substitute this back into the expression: \[ (\sqrt{49} - 2i) = 7 - 2i \]
Step 3: Substitute the simplified parts back into the expression
The expression now becomes: \[ (13 + 15i) - (7 - 2i) \]
Step 4: Distribute the negative sign
\[ (13 + 15i) - 7 + 2i = 13 - 7 + 15i + 2i \]
Step 5: Combine the real and imaginary parts
Real part: \[ 13 - 7 = 6 \] Imaginary part: \[ 15i + 2i = 17i \]
Final Result
Combining the results, we get: \[ 6 + 17i \]
Thus, the answer in the form \(a + bi\) is: \[ \boxed{6 + 17i} \]
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