Pregunta
Perform the indicated operations and express the result as a simplified complex number. \( (7-5 i)^{2} \)
Ask by Schofield Gibbs. in the United States
Mar 10,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( (7-5i)^2 = 24 - 70i \)
Solución
Calculate or simplify the expression \( (7-5i)^2 \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(7-5i\right)^{2}\)
- step1: Expand the expression:
\(7^{2}-2\times 7\times 5i+\left(5i\right)^{2}\)
- step2: Multiply:
\(7^{2}-70i-25\)
- step3: Calculate:
\(49-70i-25\)
- step4: Simplify the expression:
\(24-70i\)
The result of \( (7-5i)^{2} \) is \( 24-70i \).
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Simplifique esta solución Bonus Knowledge
To simplify \( (7-5i)^2 \), we can use the formula for squaring a binomial, \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( a = 7 \) and \( b = 5i \). Calculating each part: 1. \( a^2 = 7^2 = 49 \) 2. \( -2ab = -2(7)(5i) = -70i \) 3. \( b^2 = (5i)^2 = 25i^2 = 25(-1) = -25 \) Now, we combine these results: \[ (7 - 5i)^2 = 49 - 70i - 25 \] \[ = 24 - 70i \] So, the result is \( 24 - 70i \).
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