Question
Find the following product: \( (5-i)^{3}=\square \)
Ask by Black Guerrero. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The product of \((5-i)^{3}\) is \(110-74i\).
Solution
Calculate or simplify the expression \( (5-i)^{3} \).
Calculate or simplify the expression \( (5-i)^{3} \).
Simplify the expression by following steps:
- step0: Evaluate the power:
\(\left(5-i\right)^{3}\)
- step1: Expand the expression:
\(5^{3}-3\times 5^{2}i+3\times 5i^{2}-i^{3}\)
- step2: Evaluate:
\(125-75i-15+i\)
- step3: Simplify the expression:
\(110-74i\)
The product of \((5-i)^{3}\) is \(110-74i\).
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Simplify this solution Extra Insights
To find the product \( (5-i)^{3} \), we can first expand it using the binomial theorem or simply multiply it out step by step. 1. First, calculate \( (5-i)^{2} \): \[ (5-i)^{2} = (5-i)(5-i) = 25 - 5i - 5i + i^{2} = 25 - 10i - 1 = 24 - 10i \] 2. Now, we multiply \( (5-i) \) with \( (24 - 10i) \) to find \( (5-i)^{3} \): \[ (5-i)(24 - 10i) = 5 \cdot 24 + 5 \cdot (-10i) - i \cdot 24 - i \cdot (-10i) \] \[ = 120 - 50i - 24i + 10 = 130 - 74i \] Thus, \( (5-i)^{3} = 130 - 74i \).
