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

(14-28 i)/(1+3i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{14-28 i}{1+3 i}$ $\newline$ 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{14-28 i}{1+3 i}

\newline

Answer:

Multiply by Conjugate: To express the complex fraction $(14-28i)/(1+3i)$ in the form $a+bi$ , we need to eliminate the imaginary part 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 $(1+3i)$ is $(1-3i)$ . $\newline$ So, we multiply both the numerator and the denominator by $(1-3i)$ .
Perform Multiplication: Now, we perform the multiplication: $\newline$ $(14-28i) \times (1-3i) / (1+3i) \times (1-3i)$ $\newline$ We distribute the terms in the numerator and the denominator.
Calculate Numerator: First, we calculate the numerator: $\newline$ $(14-28i) \times (1-3i) = 14(1) - 14(3i) - 28i(1) + 28i(3i)$ $\newline$ $= 14 - 42i - 28i + 84i^2$ $\newline$ Since $i^2 = -1$ , we replace $i^2$ with $-1$ : $\newline$ $= 14 - 42i - 28i - 84$ $\newline$ $= 14 - 84 - 70i$ $\newline$ $= -70 - 70i$
Calculate Denominator: Next, we calculate the denominator: $\newline$ $(1+3i) \times (1-3i) = 1(1) - 1(3i) + 3i(1) - 3i(3i)$ $\newline$ $= 1 - 3i + 3i - 9i^2$ $\newline$ Again, since $i^2 = -1$ , we replace $i^2$ with $-1$ : $\newline$ $= 1 - 9(-1)$ $\newline$ $= 1 + 9$ $\newline$ $= 10$
Divide by Denominator: Now we have the numerator and the denominator: $\newline$ Numerator: $-70 - 70i$ $\newline$ Denominator: $10$ $\newline$ We divide both parts of the numerator by the denominator: $\newline$ $(-70 - 70i) / 10$ $\newline$ = $-70/10 - (70i/10)$ $\newline$ = $-7 - 7i$

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