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Divide the following complex numbers.

(6-7i)/(4+i)

Divide the following complex numbers. $\newline$ $\frac{6-7 i}{4+i}$

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Simplify radical expressions involving fractions

Full solution

Q. Divide the following complex numbers.

\newline

\frac{6-7 i}{4+i}

Multiply Conjugate: Multiply the numerator and denominator by the conjugate of the denominator. $\newline$ To divide complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator. The conjugate of $(4+i)$ is $(4-i)$ . $\newline$ $\frac{6-7i}{4+i} \cdot \frac{4-i}{4-i}$
Distribute Numerator: Apply the distributive property (foil method) to the numerator. $\newline$ Multiply $(6-7i)$ by $(4-i)$ . $\newline$ $(6\times 4) + (6\times (-i)) + (-7i\times 4) + (-7i\times (-i))$ $\newline$ $24 - 6i - 28i + 7i^2$ $\newline$ Since $i^2 = -1$ , replace $i^2$ with $-1$ . $\newline$ $24 - 6i - 28i - 7$ $\newline$ Combine like terms. $\newline$ $(24 - 7) + (-6i - 28i)$ $\newline$ $17 - 34i$
Distribute Denominator: Apply the distributive property to the denominator. $\newline$ Multiply $(4+i)$ by $(4-i)$ . $\newline$ $(4\cdot 4) + (4\cdot (-i)) + (i\cdot 4) + (i\cdot (-i))$ $\newline$ $16 - 4i + 4i - i^2$ $\newline$ Since $i^2 = -1$ , replace $i^2$ with $-1$ and combine like terms. $\newline$ $16 - (-1)$ $\newline$ $16 + 1$ $\newline$ $17$
Write Division Result: Write the result of the division. $\newline$ Now that we have simplified both the numerator and the denominator, we can write the result of the division. $\newline$ $(17 - 34i) / 17$
Divide Real and Imaginary: Divide both the real and imaginary parts by the denominator. $\newline$ Divide $17$ and $-34i$ by $17$ separately. $\newline$ $\frac{17}{17} - \left(\frac{34i}{17}\right)$ $\newline$ $1 - 2i$

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