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Divide the following complex numbers.

(10-20 i)/(4-3i)

Divide the following complex numbers. $\newline$ $\frac{10-20 i}{4-3 i}$

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Simplify radical expressions involving fractions

Full solution

Q. Divide the following complex numbers.

\newline

\frac{10-20 i}{4-3 i}

Find Conjugate: To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$ . So, the conjugate of $4 - 3i$ is $4 + 3i$ .
Multiply Numerators: Multiply the numerator $(10 - 20i)$ and the denominator $(4 - 3i)$ by the conjugate of the denominator $(4 + 3i)$ . $(10 - 20i) \times (4 + 3i) / (4 - 3i) \times (4 + 3i)$
Multiply Denominators: First, we'll multiply out the numerators: $\newline$ $(10 - 20i) \times (4 + 3i) = 10\times4 + 10\times3i - 20i\times4 - 20i\times3i$ $\newline$ $= 40 + 30i - 80i - 60i^2$ $\newline$ Since $i^2 = -1$ , we replace $-60i^2$ with $60$ . $\newline$ $= 40 + 30i - 80i + 60$ $\newline$ $= 100 - 50i$
Simplify Numerator and Denominator: Now, we'll multiply out the denominators: $\newline$ $(4 - 3i) \times (4 + 3i) = 4\times4 + 4\times3i - 3i\times4 - 3i\times3i$ $\newline$ $= 16 + 12i - 12i - 9i^2$ $\newline$ Again, since $i^2 = -1$ , we replace $-9i^2$ with $9$ . $\newline$ $= 16 + 12i - 12i + 9$ $\newline$ $= 16 + 9$ $\newline$ $= 25$
Divide Numerator by Denominator: Now we have the simplified numerator and denominator: $\newline$ Numerator: $100 - 50i$ $\newline$ Denominator: $25$ $\newline$ We divide the numerator by the denominator: $\newline$ $\frac{100 - 50i}{25}$
Final Simplified Form: Divide both the real part and the imaginary part of the numerator by the denominator: $\newline$ $\frac{100}{25} - \frac{50i}{25}$ $\newline$ = $4 - 2i$ $\newline$ This is the simplified form of the division of the complex numbers.

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