Pregunta
Find the product of \( -4-i \) and its conjugate
Ask by Medina Lowe. in the United States
Mar 16,2025
Solución de inteligencia artificial de Upstudy
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The product is 17.
Solución
Calculate or simplify the expression \( (-4-i) \times (-4+i) \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(-4-i\right)\left(-4+i\right)\)
- step1: Simplify the product:
\(\left(-4\right)^{2}-i^{2}\)
- step2: Evaluate the power:
\(16-\left(-1\right)\)
- step3: Calculate:
\(17\)
The product of \( -4-i \) and its conjugate is \( 17 \).
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Simplifique esta solución Mind Expander
To find the product of \( -4 - i \) and its conjugate, we first need to determine the conjugate. The conjugate of \( -4 - i \) is \( -4 + i \). Now we can multiply these two complex numbers: \[ (-4 - i)(-4 + i) \] Using the distributive property (or FOIL), we calculate: \[ (-4)(-4) + (-4)(i) + (-i)(-4) + (-i)(i) = 16 - 4i + 4i - i^2 \] The \( -4i \) and \( +4i \) cancel each other out: \[ 16 - i^2 \] Since \( i^2 = -1 \), we substitute: \[ 16 - (-1) = 16 + 1 = 17 \] Thus, the product of \( -4 - i \) and its conjugate is \( \boxed{17} \).
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