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

(4-22 i)/(2+4i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{4-22 i}{2+4 i}$ $\newline$ 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{4-22 i}{2+4 i}

\newline

Answer:

Multiply Numerators: To express the quotient $(4-22i)/(2+4i)$ as a complex number in the form $a+bi$ , 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 $(2+4i)$ is $(2-4i)$ . $\newline$ Now, multiply the numerator and the denominator by the complex conjugate of the denominator. $\newline$ $(4-22i)/(2+4i) \cdot (2-4i)/(2-4i)$
Expand Numerator: First, multiply the numerators: $(4-22i)*(2-4i)$ . Use the distributive property (FOIL method) to expand the product: $4\cdot2 + 4\cdot(-4i) - 22i\cdot2 - 22i\cdot(-4i) = 8 - 16i - 44i + 88i^2$ Since $i^2 = -1$ , replace $88i^2$ with $-88$ : $= 8 - 16i - 44i - 88$ Combine like terms: $= (8 - 88) + (-16i - 44i) = -80 - 60i$
Multiply Denominators: Next, multiply the denominators: $(2+4i)*(2-4i)$ . Again, use the distributive property to expand the product: $2\cdot2 + 2\cdot(-4i) + 4i\cdot2 - 4i\cdot4i$ = $4 - 8i + 8i - 16i^2$ Replace $-16i^2$ with $16$ , since $i^2 = -1$ : = $4 + 16$ Combine like terms: = $20$
Divide Numerator by Denominator: Now, divide the result from the numerators by the result from the denominators: $\newline$ $(-80 - 60i) / 20$ $\newline$ Divide both the real part and the imaginary part by $20$ : $\newline$ $-80/20 - (60i/20)$ $\newline$ $= -4 - 3i$
Final Complex Number: The complex number in the form $a+bi$ is $-4 - 3i$ . This is the simplest form of the given complex quotient.

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