Question

use the properties of complex numbers to simplify $(sqrt{-49} + 5i) + (8 - sqrt{-4})$. (1 point)
\bigcirc $8 + 5i + \sqrt{-53}$
\bigcirc $8 + 14i$
\bigcirc $13 + 9i$
\bigcirc $8 + 10i$

Explanation:

Step1: Simplify square roots of negative numbers

Recall that for a negative number \(-a\) (where \(a>0\)), \(\sqrt{-a}=i\sqrt{a}\). So, \(\sqrt{-49}=i\sqrt{49}=7i\) and \(\sqrt{-4}=i\sqrt{4}=2i\).

Step2: Substitute simplified roots into the expression

Substitute \(\sqrt{-49}=7i\) and \(\sqrt{-4}=2i\) into \((\sqrt{-49} + 5i)+(8-\sqrt{-4})\), we get \((7i + 5i)+(8 - 2i)\)? Wait, no, wait: the original expression is \((\sqrt{-49}+5i)+(8 - \sqrt{-4})\), so substituting gives \((7i + 5i)+8 - 2i\)? Wait, no, \(\sqrt{-49}=7i\), so the first part is \(7i + 5i\), and the second part is \(8 - 2i\) (since \(\sqrt{-4}=2i\), so \(-\sqrt{-4}=-2i\)). Wait, no, let's re - write the original expression: \((\sqrt{-49}+5i)+(8-\sqrt{-4})=(7i + 5i)+8-2i\)? Wait, no, \(\sqrt{-49}=i\sqrt{49} = 7i\), so the first term inside the first parentheses is \(7i\), then plus \(5i\), so \(7i+5i = 12i\)? Wait, no, I made a mistake. Wait, \(\sqrt{-49}=i\sqrt{49}=7i\), so the first part is \(7i + 5i\)? Wait, no, the original expression is \((\sqrt{-49}+5i)+(8-\sqrt{-4})\). So \(\sqrt{-49}=7i\), so the first parentheses is \(7i + 5i\), and the second parentheses is \(8-2i\) (because \(\sqrt{-4}=2i\), so \(-\sqrt{-4}=-2i\)). Wait, no, let's do it step by step:

\((\sqrt{-49}+5i)+(8-\sqrt{-4})=(i\sqrt{49}+5i)+(8 - i\sqrt{4})\)

\(=(7i + 5i)+(8-2i)\)

Now, combine like terms. The real part is just \(8\). The imaginary parts: \(7i+5i-2i=(7 + 5-2)i=10i\)? Wait, no, wait, I messed up the sign. Wait, the second term is \(-\sqrt{-4}\), which is \(-2i\), so the expression is \((7i + 5i)+8-2i\)? No, the original expression is \((\sqrt{-49}+5i)+(8-\sqrt{-4})\), so it's \(7i+5i + 8-2i\). Now, combine the imaginary parts: \(7i+5i-2i=(7 + 5-2)i = 10i\)? Wait, no, that's not right. Wait, \(7i+5i=12i\), then \(12i-2i = 10i\), and the real part is \(8\). Wait, but that's not one of the options. Wait, I must have made a mistake. Wait, \(\sqrt{-49}=7i\), so the first term is \(7i\), then plus \(5i\) gives \(7i + 5i=12i\), and then we have \(8-\sqrt{-4}=8 - 2i\). So adding \((12i)+(8 - 2i)=8+(12i-2i)=8 + 10i\)? Wait, no, the options have \(8 + 10i\) as one of the options. Wait, let's re - calculate:

\(\sqrt{-49}=i\sqrt{49}=7i\)

\(\sqrt{-4}=i\sqrt{4}=2i\)

So the expression \((\sqrt{-49}+5i)+(8-\sqrt{-4})\) becomes:

\((7i + 5i)+8-2i\)? No, wait, the first parentheses is \(\sqrt{-49}+5i=7i + 5i\), and the second parentheses is \(8-\sqrt{-4}=8 - 2i\). Now, add the two parentheses together: \((7i + 5i)+(8-2i)=7i+5i + 8-2i\). Combine like terms:

Real part: \(8\)

Imaginary part: \(7i+5i-2i=(7 + 5-2)i=10i\)

So the simplified expression is \(8 + 10i\)

Answer:

\(8 + 10i\) (the option with \(8 + 10i\))