Question
Use the properties of complex numbers to simplify (−49−−−−√+5i)+(8−(−4)−−−−√)
(1 point)
Responses
8+5i+−53−−−−√
8 plus 5 i plus square root of negative 53
8+10i
8 plus 10 i
13+9i
13 plus 9 i
8+14i
(1 point)
Responses
8+5i+−53−−−−√
8 plus 5 i plus square root of negative 53
8+10i
8 plus 10 i
13+9i
13 plus 9 i
8+14i
Answers
Answered by
GPT-4o mini
To simplify the given expression \((-\sqrt{49}+5i)+(8-\sqrt{-4})\), let's break it down step by step.
1. Simplify \(-\sqrt{49}\):
\[
-\sqrt{49} = -7
\]
2. Simplify \(\sqrt{-4}\):
\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i
\]
3. Substitute these values back into the expression:
\[
(-7 + 5i) + (8 - 2i)
\]
4. Combine the real parts and the imaginary parts:
- Real parts: \(-7 + 8 = 1\)
- Imaginary parts: \(5i - 2i = 3i\)
5. So, the simplified expression is:
\[
1 + 3i
\]
Therefore, the final simplified form of the expression is \(1 + 3i\). However, it seems this answer isn't one of the provided options. Let's check the calculations again:
1. Original expression:
\[
(-\sqrt{49}+5i)+(8-\sqrt{-4})
\]
2. Simplifying:
- \(-\sqrt{49} = -7\)
- \(\sqrt{-4} = 2i\)
3. So the expression becomes:
\[
(-7 + 5i) + (8 - 2i) = 1 + 3i
\]
The answer does not match any of the options provided:
- \(8+5i+\sqrt{-53}\)
- \(8+10i\)
- \(13+9i\)
- \(8+14i\)
Therefore, it appears that the options do not correspond to the correct simplification given the problem. The accurate simplified form is \(1 + 3i\), which is not listed in the responses. Please double-check the original problem for any possible errors in formulation or questioning.
1. Simplify \(-\sqrt{49}\):
\[
-\sqrt{49} = -7
\]
2. Simplify \(\sqrt{-4}\):
\[
\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i
\]
3. Substitute these values back into the expression:
\[
(-7 + 5i) + (8 - 2i)
\]
4. Combine the real parts and the imaginary parts:
- Real parts: \(-7 + 8 = 1\)
- Imaginary parts: \(5i - 2i = 3i\)
5. So, the simplified expression is:
\[
1 + 3i
\]
Therefore, the final simplified form of the expression is \(1 + 3i\). However, it seems this answer isn't one of the provided options. Let's check the calculations again:
1. Original expression:
\[
(-\sqrt{49}+5i)+(8-\sqrt{-4})
\]
2. Simplifying:
- \(-\sqrt{49} = -7\)
- \(\sqrt{-4} = 2i\)
3. So the expression becomes:
\[
(-7 + 5i) + (8 - 2i) = 1 + 3i
\]
The answer does not match any of the options provided:
- \(8+5i+\sqrt{-53}\)
- \(8+10i\)
- \(13+9i\)
- \(8+14i\)
Therefore, it appears that the options do not correspond to the correct simplification given the problem. The accurate simplified form is \(1 + 3i\), which is not listed in the responses. Please double-check the original problem for any possible errors in formulation or questioning.