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

(-9-6i)/(-5+2i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{-9-6 i}{-5+2 i}$ $\newline$ 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{-9-6 i}{-5+2 i}

\newline

Answer:

Multiply by Conjugate: To simplify the complex fraction $(-9-6i)/(-5+2i)$ , we need to eliminate the imaginary part from the denominator. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(-5+2i)$ is $(-5-2i)$ .
Expand Numerator: Now, we multiply the numerator and the denominator by the conjugate of the denominator: $((-9-6i) \times (-5-2i)) / ((-5+2i) \times (-5-2i))$ .
Calculate Numerator: First, we'll expand the numerator: $\newline$ $(-9-6i) * (-5-2i) = (-9 * -5) + (-9 * -2i) + (-6i * -5) + (-6i * -2i)$ .
Combine Like Terms: Now, we calculate the products in the numerator: $\newline$ $45 + 18i + 30i + 12i^2$ . $\newline$ Since $i^2 = -1$ , we replace $12i^2$ with $-12$ : $\newline$ $45 + 18i + 30i - 12$ .
Expand Denominator: Combine like terms in the numerator: $\newline$ $(45 - 12) + (18i + 30i) = 33 + 48i$ .
Calculate Denominator: Next, we'll expand the denominator: $\newline$ $(-5+2i) * (-5-2i) = (-5 * -5) + (-5 * -2i) + (2i * -5) + (2i * -2i)$ .
Combine Like Terms: Now, we calculate the products in the denominator: $25 - 10i - 10i + 4i^2$ . Again, since $i^2 = -1$ , we replace $4i^2$ with $-4$ : $25 - 10i - 10i - 4$ .
Simplify Numerator and Denominator: Combine like terms in the denominator: \(25 - $4$ ) - ( $10$ i - $10$ i) = $21$ \
Divide Numerator by Denominator: Now we have the simplified numerator and denominator: $\newline$ Numerator: $33 + 48i$ $\newline$ Denominator: $21$ $\newline$ We divide both parts of the numerator by the denominator to get the complex number in $a+bi$ form: $\newline$ $(\frac{33}{21}) + (\frac{48i}{21})$ .
Simplify Complex Number: Simplify both parts of the complex number: $\frac{33}{21}$ simplifies to $\frac{11}{7}$ and $\frac{48}{21}$ simplifies to $\frac{16}{7}$ . So, the complex number in $a+bi$ form is: $(\frac{11}{7}) + (\frac{16}{7})i$ .

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