Question
Perform the indicated operations. Write the answer in standard form, \( a+b i \). \[ (2-6 i)^{2}+(2+6 i)^{2}=\square \]
Ask by Vaughan Dunn. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The answer is \(-64\).
Solution
Calculate or simplify the expression \( (2-6i)^2+(2+6i)^2 \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(2-6i\right)^{2}+\left(2+6i\right)^{2}\)
- step1: Evaluate the power:
\(-32-24i+\left(2+6i\right)^{2}\)
- step2: Evaluate the power:
\(-32-24i-32+24i\)
- step3: Add the numbers:
\(-64\)
The result of the expression \((2-6i)^2+(2+6i)^2\) is \(-64\).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
Let's start by expanding both expressions in the equation. First, we calculate \((2-6i)^{2}\): \[ (2 - 6i)^{2} = 2^{2} - 2 \cdot 2 \cdot 6i + (6i)^{2} = 4 - 24i - 36 = -32 - 24i \] Next, we calculate \((2+6i)^{2}\): \[ (2 + 6i)^{2} = 2^{2} + 2 \cdot 2 \cdot 6i + (6i)^{2} = 4 + 24i - 36 = -32 + 24i \] Now, we can add these two results: \[ (2 - 6i)^{2} + (2 + 6i)^{2} = (-32 - 24i) + (-32 + 24i) \] When combining like terms: \[ -32 - 32 + (-24i + 24i) = -64 + 0i \] Thus, the final answer in standard form \( a + bi \) is: \[ \boxed{-64} \]
