We read your photo as:

Add the real parts and the imaginary parts.

Mathematics

ScanSolve AI Answer

Step-by-step solution

23 steps

Let z1=−3+2iz_1 = -3 + 2i and z2=4−iz_2 = 4 - i.

Part i· 1 step

z1+z2z_1 + z_2

  1. Add the real parts and the imaginary parts.

    z1+z2=(−3+2i)+(4−i)z_1 + z_2 = (-3 + 2i) + (4 - i) =(−3+4)+(2−1)i= (-3 + 4) + (2 - 1)i =1+i= 1 + i

    The result is 1+i\boxed{1+i}.

Part ii· 1 step

z1−z2z_1 - z_2

  1. Subtract the real parts and the imaginary parts.

    z1−z2=(−3+2i)−(4−i)z_1 - z_2 = (-3 + 2i) - (4 - i) =−3+2i−4+i= -3 + 2i - 4 + i =(−3−4)+(2+1)i= (-3 - 4) + (2 + 1)i =−7+3i= -7 + 3i

    The result is −7+3i\boxed{-7+3i}.

Part iii· 2 steps

z1z2z_1 z_2

  1. Multiply the complex numbers using the distributive property.

    z1z2=(−3+2i)(4−i)z_1 z_2 = (-3 + 2i)(4 - i) =−3(4)+(−3)(−i)+2i(4)+2i(−i)= -3(4) + (-3)(-i) + 2i(4) + 2i(-i) =−12+3i+8i−2i2= -12 + 3i + 8i - 2i^2

  2. Substitute i2=−1i^2 = -1 and simplify.

    =−12+11i−2(−1)= -12 + 11i - 2(-1) =−12+11i+2= -12 + 11i + 2 =−10+11i= -10 + 11i

    The result is −10+11i\boxed{-10+11i}.

Part iv· 4 steps

z1z2\frac{z_1}{z_2}

  1. Multiply the numerator and denominator by the conjugate of z2z_2.

    The conjugate of z2=4−iz_2 = 4-i is 4+i4+i.

    z1z2=−3+2i4−i×4+i4+i\frac{z_1}{z_2} = \frac{-3 + 2i}{4 - i} \times \frac{4 + i}{4 + i}

  2. Expand the numerator.

    (−3+2i)(4+i)=−12−3i+8i+2i2(-3 + 2i)(4 + i) = -12 - 3i + 8i + 2i^2 =−12+5i+2(−1)=−14+5i= -12 + 5i + 2(-1) = -14 + 5i

  3. Expand the denominator.

    (4−i)(4+i)=42−i2(4 - i)(4 + i) = 4^2 - i^2 =16−(−1)=17= 16 - (-1) = 17

  4. Combine and express in Cartesian form.

    −14+5i17=−1417+517i\frac{-14 + 5i}{17} = -\frac{14}{17} + \frac{5}{17}i

    The result is −1417+517i\boxed{-\frac{14}{17} + \frac{5}{17}i}.

Part v· 2 steps

2z1−5z22z_1 - 5z_2

  1. Multiply z1z_1 by 2 and z2z_2 by 5.

    2z1=2(−3+2i)=−6+4i2z_1 = 2(-3 + 2i) = -6 + 4i 5z2=5(4−i)=20−5i5z_2 = 5(4 - i) = 20 - 5i

  2. Subtract the results.

    2z1−5z2=(−6+4i)−(20−5i)2z_1 - 5z_2 = (-6 + 4i) - (20 - 5i) =−6+4i−20+5i= -6 + 4i - 20 + 5i =(−6−20)+(4+5)i= (-6 - 20) + (4 + 5)i =−26+9i= -26 + 9i

    The result is −26+9i\boxed{-26+9i}.

Part vi· 2 steps

−z1−2z2-z_1 - 2z_2

  1. Multiply z1z_1 by −1-1 and z2z_2 by −2-2.

    −z1=−(−3+2i)=3−2i-z_1 = -(-3 + 2i) = 3 - 2i −2z2=−2(4−i)=−8+2i-2z_2 = -2(4 - i) = -8 + 2i

  2. Add the results.

    −z1−2z2=(3−2i)+(−8+2i)-z_1 - 2z_2 = (3 - 2i) + (-8 + 2i) =(3−8)+(−2+2)i= (3 - 8) + (-2 + 2)i =−5+0i=−5= -5 + 0i = -5

    The result is −5\boxed{-5}.

Part vii· 2 steps

(−z1)(−z2)(-z_1)(-z_2)

  1. Multiply −z1-z_1 and −z2-z_2.

    This is equivalent to z1z2z_1 z_2.

    (−z1)(−z2)=(−(−3+2i))(−(4−i))(-z_1)(-z_2) = (-(-3 + 2i))(-(4 - i)) =(3−2i)(−4+i)= (3 - 2i)(-4 + i) =3(−4)+3(i)−2i(−4)−2i(i)= 3(-4) + 3(i) - 2i(-4) - 2i(i) =−12+3i+8i−2i2= -12 + 3i + 8i - 2i^2

  2. Substitute i2=−1i^2 = -1 and simplify.

    =−12+11i−2(−1)= -12 + 11i - 2(-1) =−12+11i+2= -12 + 11i + 2 =−10+11i= -10 + 11i

    The result is −10+11i\boxed{-10+11i}.

Part viii· 2 steps

3z1+2z23z_1 + 2z_2

  1. Multiply z1z_1 by 3 and z2z_2 by 2.

    3z1=3(−3+2i)=−9+6i3z_1 = 3(-3 + 2i) = -9 + 6i 2z2=2(4−i)=8−2i2z_2 = 2(4 - i) = 8 - 2i

  2. Add the results.

    3z1+2z2=(−9+6i)+(8−2i)3z_1 + 2z_2 = (-9 + 6i) + (8 - 2i) =(−9+8)+(6−2)i= (-9 + 8) + (6 - 2)i =−1+4i= -1 + 4i

    The result is −1+4i\boxed{-1+4i}.

Part ix· 3 steps

i(z1+z2)i(z_1 + z_2)

  1. Calculate z1+z2z_1 + z_2.

    z1+z2=(−3+2i)+(4−i)=1+iz_1 + z_2 = (-3 + 2i) + (4 - i) = 1 + i

  2. Multiply the sum by ii.

    i(z1+z2)=i(1+i)i(z_1 + z_2) = i(1 + i) =i(1)+i(i)= i(1) + i(i) =i+i2= i + i^2

  3. Substitute i2=−1i^2 = -1 and simplify.

    =i−1=−1+i= i - 1 = -1 + i

    The result is −1+i\boxed{-1+i}.

Part x· 4 steps

z2z1\frac{z_2}{z_1}

  1. Multiply the numerator and denominator by the conjugate of z1z_1.

    The conjugate of z1=−3+2iz_1 = -3+2i is −3−2i-3-2i.

    z2z1=4−i−3+2i×−3−2i−3−2i\frac{z_2}{z_1} = \frac{4 - i}{-3 + 2i} \times \frac{-3 - 2i}{-3 - 2i}

  2. Expand the numerator.

    (4−i)(−3−2i)=4(−3)+4(−2i)−i(−3)−i(−2i)(4 - i)(-3 - 2i) = 4(-3) + 4(-2i) - i(-3) - i(-2i) =−12−8i+3i+2i2= -12 - 8i + 3i + 2i^2 =−12−5i+2(−1)=−14−5i= -12 - 5i + 2(-1) = -14 - 5i

  3. Expand the denominator.

    (−3+2i)(−3−2i)=(−3)2−(2i)2(-3 + 2i)(-3 - 2i) = (-3)^2 - (2i)^2 =9−4i2= 9 - 4i^2 =9−4(−1)=13= 9 - 4(-1) = 13

  4. Combine and express in Cartesian form.

    −14−5i13=−1413−513i\frac{-14 - 5i}{13} = -\frac{14}{13} - \frac{5}{13}i

    The result is −1413−513i\boxed{-\frac{14}{13} - \frac{5}{13}i}.

Was this helpful?

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QAdd the complex numbers (3+i4) and (−5-i8).QWhat are the real and imaginary parts of the following complex numbers: i) −3 − 4i, ii) 20 + 4i…QMultiply the complex numbers z1 = 2+3i and z2 = −3+i, and find the real part of the product.QExpress the following complex numbers in Cartesian form, given their real (x) and imaginary (y)…QSubtract the complex numbers (2, 5) and (3, −2). Is the imaginary part of the first number −y?
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp