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Express as a complex number in simplest a+bi form:

(10-5i)/(-5-6i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{10-5 i}{-5-6 i}$ $\newline$ Answer:

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Even and odd functions

Full solution

Q. Express as a complex number in simplest a+bi form:

\newline

\frac{10-5 i}{-5-6 i}

\newline

Answer:

Identify Conjugate: To simplify a complex fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$ . So, the conjugate of $-5 - 6i$ is $-5 + 6i$ .
Multiply by Conjugate: Now, we multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. $\newline$ $(10 - 5i) \times (-5 + 6i) / (-5 - 6i) \times (-5 + 6i)$
Distribute Multiplication: We distribute the multiplication in the numerator and the denominator. $\newline$ Numerator: $10 \times -5$ + $10 \times 6i$ + $-5i \times -5$ + $-5i \times 6i$ $\newline$ Denominator: $-5 \times -5$ + $-5 \times 6i$ + $-6i \times -5$ + $-6i \times 6i$
Simplify Multiplication: We simplify the multiplication. $\newline$ Numerator: $-50 + 60i + 25i - 30i^2$ $\newline$ Denominator: $25 - 30i + 30i - 36i^2$ $\newline$ Note that $i^2 = -1$ .
Replace $i^2$ : We replace $i^2$ with $-1$ and simplify further. $\newline$ Numerator: $-50 + 60i + 25i + 30$ $\newline$ Denominator: $25 + 36$ $\newline$ Numerator: $-20 + 85i$ $\newline$ Denominator: $61$
Divide by Denominator: We divide the real and imaginary parts of the numerator by the denominator to get the complex number in $a+bi$ form. $\newline$ Real part: $-\frac{20}{61}$ $\newline$ Imaginary part: $\frac{85i}{61}$
Final Answer: We write the final answer in $a+bi$ form. $\newline$ Final answer: $\left(-\frac{20}{61}\right) + \left(\frac{85}{61}\right)i$

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