Question

find the sum or difference.
$-i\sqrt{2} - 5 - (5 - 6i\sqrt{2}) - (5 - i\sqrt{2})$

$-i\sqrt{2} - 5 - (5 - 6i\sqrt{2}) - (5 - i\sqrt{2}) = \square$
(simplify your answer. type your answer in the form $a + bi$. type an

Explanation:

Step1: Remove parentheses

First, we distribute the negative signs to the terms inside the parentheses. Remember that \( - (a - b) = -a + b \).
So, \( -i\sqrt{2} - 5 - (5 - 6i\sqrt{2}) - (5 - i\sqrt{2}) = -i\sqrt{2} - 5 - 5 + 6i\sqrt{2} - 5 + i\sqrt{2} \)

Step2: Combine real parts

Now, we combine the real parts (the terms without \( i \)). The real parts are \( -5 - 5 - 5 \).
\( -5 - 5 - 5 = -15 \)

Step3: Combine imaginary parts

Next, we combine the imaginary parts (the terms with \( i \)). The imaginary parts are \( -i\sqrt{2} + 6i\sqrt{2} + i\sqrt{2} \).
Factor out \( i\sqrt{2} \): \( i\sqrt{2}(-1 + 6 + 1) = i\sqrt{2}(6) = 6i\sqrt{2} \)

Step4: Combine real and imaginary parts

Now, we combine the real part and the imaginary part to get the result in the form \( a + bi \).
So, the real part \( a = -15 \) and the imaginary part \( b = 6\sqrt{2} \), so the expression simplifies to \( -15 + 6\sqrt{2}i \)

Answer:

\( -15 + 6\sqrt{2}i \)