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

(9-2i)/(-7-3i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{9-2 i}{-7-3 i}$ $\newline$ 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{9-2 i}{-7-3 i}

\newline

Answer:

Multiply by Conjugate: Multiply the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part in the denominator. $\newline$ The complex conjugate of $(-7-3i)$ is $(-7+3i)$ . $\newline$ $\frac{9-2i}{-7-3i} \cdot \frac{-7+3i}{-7+3i}$
Apply Distributive Property: Apply the distributive property (foil method) to multiply out the numerators and the denominators. $\newline$ Numerator: $(9-2i)(-7+3i) = 9\cdot(-7) + 9\cdot3i + (-2i)\cdot(-7) + (-2i)\cdot3i$ $\newline$ Denominator: $(-7-3i)(-7+3i) = (-7)\cdot(-7) + (-7)\cdot3i + (-3i)\cdot(-7) - 3i\cdot3i$
Perform Multiplication: Perform the multiplication. $\newline$ Numerator: $-63 + 27i - 14i - 6i^2$ $\newline$ Denominator: $49 - 21i + 21i - 9i^2$ $\newline$ Since $i^2 = -1$ , replace $i^2$ with $-1$ in both the numerator and the denominator.
Simplify Using $i^2$ : Simplify the expressions by combining like terms and using $i^2 = -1$ . $\newline$ Numerator: $-63 + 27i - 14i + 6(-1) = -63 + 13i - 6 = -69 + 13i$ $\newline$ Denominator: $49 - 21i + 21i + 9(-1) = 49 + 9 = 58$
Divide Numerator by Denominator: Divide the simplified numerator by the simplified denominator. $\newline$ $(-69 + 13i) / 58$ $\newline$ Separate the real and imaginary parts and divide each by $58$ . $\newline$ Real part: $-69/58$ $\newline$ Imaginary part: $13i/58$
Simplify Fractions: Simplify the fractions. $\newline$ Real part: $-\frac{69}{58} = -1.18965517241$ (approximately $-1.19$ ) $\newline$ Imaginary part: $\frac{13i}{58} = 0.22413793103i$ (approximately $0.22i$ )
Final Answer: Write the final answer in $a+bi$ form. $\newline$ $-1.19 + 0.22i$

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