Solved on Jan 24, 2024

Divide the complex number $\frac{-7-j 5}{-4+j 6}$ and write the result in $a+j b$ form. Round the final answers to 4 decimal places.
$a=$ $b=$

STEP 1

Assumptions
1. We are given a complex division problem: $\frac{-7-j 5}{-4+j 6}$ .
2. We need to express the answer in the form $a+j b$ , where $a$ and $b$ are real numbers.
3. The imaginary unit is denoted by $j$ , which is equivalent to $i$ in mathematics, and $j^2 = -1$ .

STEP 2

To divide complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a complex number $a + j b$ is $a - j b$ .

STEP 3

Find the complex conjugate of the denominator $-4+j 6$ .
$Complex\, conjugate = -4 - j 6$

STEP 4

Multiply the numerator and denominator of the given fraction by the complex conjugate of the denominator.
$\frac{-7-j 5}{-4+j 6} \cdot \frac{-4-j 6}{-4-j 6}$

STEP 5

Apply the distributive property to multiply out the numerators.
$Numerator = (-7-j 5)(-4-j 6)$

STEP 6

Expand the numerator using the FOIL method (First, Outer, Inner, Last).
$Numerator = (-7 \cdot -4) + (-7 \cdot -j 6) + (-j 5 \cdot -4) + (-j 5 \cdot -j 6)$

STEP 7

Calculate each term in the expansion.
$Numerator = 28 + 42j + 20j + 30j^2$

STEP 8

Since $j^2 = -1$ , replace $30j^2$ with $-30$ .
$Numerator = 28 + 42j + 20j - 30$

STEP 9

Combine like terms in the numerator.
$Numerator = (28 - 30) + (42j + 20j)$

STEP 10

Simplify the real and imaginary parts of the numerator.
$Numerator = -2 + 62j$

STEP 11

Apply the distributive property to multiply out the denominators.
$Denominator = (-4 - j 6)(-4 + j 6)$

STEP 12

Expand the denominator using the difference of squares formula, $(a - b)(a + b) = a^2 - b^2$ .
$Denominator = (-4)^2 - (j 6)^2$

STEP 13

Calculate each term in the expansion.
$Denominator = 16 - 36j^2$

STEP 14

Since $j^2 = -1$ , replace $36j^2$ with $-36(-1)$ .
$Denominator = 16 - (-36)$

STEP 15

Simplify the denominator.
$Denominator = 16 + 36$

STEP 16

Calculate the simplified denominator.
$Denominator = 52$

STEP 17

Now that we have the simplified numerator and denominator, we can write the division as a fraction.
$\frac{-2 + 62j}{52}$

STEP 18

Separate the real and imaginary parts of the fraction.
$\frac{-2}{52} + \frac{62j}{52}$

STEP 19

Simplify the real part of the fraction.
$a = \frac{-2}{52}$

STEP 20

Simplify the imaginary part of the fraction.
$b = \frac{62}{52}$

STEP 21

Reduce the fractions to their simplest form.
$a = \frac{-1}{26}$
$b = \frac{31}{26}$

STEP 22

Convert the simplified fractions to decimal form, if necessary, rounding to four decimal places.
$a \approx -0.0385$
$b \approx 1.1923$

STEP 23

Write the final answer in $a+j b$ form.
$a+j b = -0.0385 + j 1.1923$
Therefore, the division of the given complex numbers is:
$\begin{array}{l} a= -0.0385 \\ b= 1.1923 \end{array}$

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Divide the complex number −7−j5−4+j6\frac{-7-j 5}{-4+j 6}−4+j6−7−j5​ and write the result in a+jba+j ba+jb form. Round the final answers to 4 decimal places.a=a=a= b=b=b=

STEP 1

STEP 2

To divide complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a+jba + j ba+jb is a−jba - j ba−jb.

STEP 3

Find the complex conjugate of the denominator −4+j6-4+j 6−4+j6.Complex conjugate=−4−j6Complex\, conjugate = -4 - j 6Complexconjugate=−4−j6

STEP 4

Multiply the numerator and denominator of the given fraction by the complex conjugate of the denominator.−7−j5−4+j6⋅−4−j6−4−j6\frac{-7-j 5}{-4+j 6} \cdot \frac{-4-j 6}{-4-j 6}−4+j6−7−j5​⋅−4−j6−4−j6​

STEP 5

Apply the distributive property to multiply out the numerators.Numerator=(−7−j5)(−4−j6)Numerator = (-7-j 5)(-4-j 6)Numerator=(−7−j5)(−4−j6)

STEP 6

Expand the numerator using the FOIL method (First, Outer, Inner, Last).Numerator=(−7⋅−4)+(−7⋅−j6)+(−j5⋅−4)+(−j5⋅−j6)Numerator = (-7 \cdot -4) + (-7 \cdot -j 6) + (-j 5 \cdot -4) + (-j 5 \cdot -j 6)Numerator=(−7⋅−4)+(−7⋅−j6)+(−j5⋅−4)+(−j5⋅−j6)

STEP 7

Calculate each term in the expansion.Numerator=28+42j+20j+30j2Numerator = 28 + 42j + 20j + 30j^2Numerator=28+42j+20j+30j2

STEP 8

Since j2=−1j^2 = -1j2=−1, replace 30j230j^230j2 with −30-30−30.Numerator=28+42j+20j−30Numerator = 28 + 42j + 20j - 30Numerator=28+42j+20j−30

STEP 9

Combine like terms in the numerator.Numerator=(28−30)+(42j+20j)Numerator = (28 - 30) + (42j + 20j)Numerator=(28−30)+(42j+20j)

STEP 10

Simplify the real and imaginary parts of the numerator.Numerator=−2+62jNumerator = -2 + 62jNumerator=−2+62j

STEP 11

Apply the distributive property to multiply out the denominators.Denominator=(−4−j6)(−4+j6)Denominator = (-4 - j 6)(-4 + j 6)Denominator=(−4−j6)(−4+j6)

STEP 12

Expand the denominator using the difference of squares formula, (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)(a+b)=a2−b2.Denominator=(−4)2−(j6)2Denominator = (-4)^2 - (j 6)^2Denominator=(−4)2−(j6)2

STEP 13

Calculate each term in the expansion.Denominator=16−36j2Denominator = 16 - 36j^2Denominator=16−36j2

STEP 14

Since j2=−1j^2 = -1j2=−1, replace 36j236j^236j2 with −36(−1)-36(-1)−36(−1).Denominator=16−(−36)Denominator = 16 - (-36)Denominator=16−(−36)

STEP 15

Simplify the denominator.Denominator=16+36Denominator = 16 + 36Denominator=16+36

STEP 16

Calculate the simplified denominator.Denominator=52Denominator = 52Denominator=52

STEP 17

Now that we have the simplified numerator and denominator, we can write the division as a fraction.−2+62j52\frac{-2 + 62j}{52}52−2+62j​

STEP 18

Separate the real and imaginary parts of the fraction.−252+62j52\frac{-2}{52} + \frac{62j}{52}52−2​+5262j​

STEP 19

Simplify the real part of the fraction.a=−252a = \frac{-2}{52}a=52−2​

STEP 20

Simplify the imaginary part of the fraction.b=6252b = \frac{62}{52}b=5262​

STEP 21

Reduce the fractions to their simplest form.a=−126a = \frac{-1}{26}a=26−1​b=3126b = \frac{31}{26}b=2631​

STEP 22

Convert the simplified fractions to decimal form, if necessary, rounding to four decimal places.a≈−0.0385a \approx -0.0385a≈−0.0385b≈1.1923b \approx 1.1923b≈1.1923

STEP 23