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

(-19-i)/(10-9i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{-19-i}{10-9 i}$ $\newline$ Answer:

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Convert complex numbers between rectangular and polar form

Full solution

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

\newline

\frac{-19-i}{10-9 i}

\newline

Answer:

Multiply by Conjugate: To express the complex number $(-19-i)/(10-9i)$ in simplest $a+bi$ form, we need to eliminate the imaginary unit $i$ from the denominator. We do this by multiplying the numerator and the denominator by the complex conjugate of the denominator. $\newline$ The complex conjugate of $(10-9i)$ is $(10+9i)$ .
Expand Numerator: Now, we multiply both the numerator and the denominator by the complex conjugate $(10+9i)$ : $\newline$ $((-19-i) \cdot (10+9i)) / ((10-9i) \cdot (10+9i))$
Expand Denominator: We expand the numerator: $\newline$ $(-19 \times 10) + (-19 \times 9i) + (-i \times 10) + (-i \times 9i)$ $\newline$ $= -190 - 171i - 10i + 9$ $\newline$ $= -181 - 181i$
Divide Numerator by Denominator: We expand the denominator: $\newline$ $(10 \times 10) + (10 \times 9i) + (-9i \times 10) + (-9i \times 9i)$ $\newline$ $= 100 + 90i - 90i - 81i^2$ $\newline$ Since $i^2 = -1$ , we have: $\newline$ $= 100 - 81(-1)$ $\newline$ $= 100 + 81$ $\newline$ $= 181$
Separate Real and Imaginary Parts: Now we divide the simplified numerator by the simplified denominator: $(-181 - 181i) / 181$
Combine Real and Imaginary Parts: We separate the real and imaginary parts and divide each by $181$ :
Real part: $-181 / 181 = -1$
Imaginary part: $-181i / 181 = -i$
Combine Real and Imaginary Parts: We separate the real and imaginary parts and divide each by $181$ : $\newline$ Real part: $-181 / 181 = -1$ $\newline$ Imaginary part: $-181i / 181 = -i$ Combining the real and imaginary parts, we get the complex number in $a+bi$ form: $\newline$ $-1 - i$

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