Divide the following complex numbers. (6-7i)/(4+i)

Multiply Conjugate: Multiply the numerator and denominator by the conjugate of the denominator. 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). (6-7i)/(4+i) * (4-i)/(4-i)Distribute Numerator: Apply the distributive property (foil method) to the numerator. Multiply (6-7i) by (4-i). (6*4) + (6*(-i)) + (-7i*4) + (-7i*(-i)) 24 - 6i - 28i + 7i^2 Since i^2 = -1, replace i^2 with -1. 24 - 6i - 28i - 7 Combine like terms. (24 - 7) + (-6i - 28i) 17 - 34iDistribute Denominator: Apply the distributive property to the denominator. Multiply (4+i) by (4-i). (4*4) + (4*(-i)) + (i*4) + (i*(-i)) 16 - 4i + 4i - i^2 Since i^2 = -1, replace i^2 with -1 and combine like terms. 16 - (-1) 16 + 1 17Write Division Result: Write the result of the division. Now that we have simplified both the numerator and the denominator, we can write the result of the division. (17 - 34i) / 17Divide Real and Imaginary: Divide both the real and imaginary parts by the denominator. Divide 17 and -34i by 17 separately. 17/17 - (34i/17) 1 - 2i

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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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Algebra 1

Simplify radical expressions involving fractions

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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. $\newline$ $\frac{17}{17} - \left(\frac{34i}{17}\right)$ $\newline$ $1 - 2i$