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Simplify the following: 4/i (1-i)/(1+i) (4+5i)/(6-5i) 4i/((1+2i)2)

Mathematics

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Step-by-step solution

13 steps

Here are the simplified forms of the complex numbers:

Part a· 2 steps

Simplify 4i\frac{4}{i}

  1. Multiply the numerator and denominator by ii to eliminate ii from the denominator.

    4i=4i×ii\frac{4}{i} = \frac{4}{i} \times \frac{i}{i}

  2. Perform the multiplication and use the identity i2=−1i^2 = -1.

    4ii2=4i−1=−4i\frac{4i}{i^2} = \frac{4i}{-1} = -4i

    The simplified form is -4i\boxed{\text{-4i}}.

Part b· 3 steps

Simplify 1−i1+i\frac{1-i}{1+i}

  1. Multiply the numerator and denominator by the conjugate of the denominator, which is 1−i1-i.

    1−i1+i=1−i1+i×1−i1−i\frac{1-i}{1+i} = \frac{1-i}{1+i} \times \frac{1-i}{1-i}

  2. Expand the numerator and the denominator.

    Numerator: (1−i)(1−i)=1−i−i+i2=1−2i+(−1)=−2i(1-i)(1-i) = 1 - i - i + i^2 = 1 - 2i + (-1) = -2i
    Denominator: (1+i)(1−i)=12−i2=1−(−1)=1+1=2(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2

  3. Combine the simplified numerator and denominator.

    −2i2=−i\frac{-2i}{2} = -i

    The simplified form is -i\boxed{\text{-i}}.

Part c· 3 steps

Simplify 4+5i6−5i\frac{4+5i}{6-5i}

  1. Multiply the numerator and denominator by the conjugate of the denominator, which is 6+5i6+5i.

    4+5i6−5i=4+5i6−5i×6+5i6+5i\frac{4+5i}{6-5i} = \frac{4+5i}{6-5i} \times \frac{6+5i}{6+5i}

  2. Expand the numerator and the denominator.

    Numerator: (4+5i)(6+5i)=4(6)+4(5i)+5i(6)+5i(5i)=24+20i+30i+25i2=24+50i+25(−1)=24+50i−25=−1+50i(4+5i)(6+5i) = 4(6) + 4(5i) + 5i(6) + 5i(5i) = 24 + 20i + 30i + 25i^2 = 24 + 50i + 25(-1) = 24 + 50i - 25 = -1 + 50i
    Denominator: (6−5i)(6+5i)=62−(5i)2=36−25i2=36−25(−1)=36+25=61(6-5i)(6+5i) = 6^2 - (5i)^2 = 36 - 25i^2 = 36 - 25(-1) = 36 + 25 = 61

  3. Combine the simplified numerator and denominator.

    −1+50i61=−161+5061i\frac{-1+50i}{61} = -\frac{1}{61} + \frac{50}{61}i

    The simplified form is \boxed{\text{-\frac{1}{61} + \frac{50}{61}i}}.

Part d· 5 steps

Simplify 4i(1+2i)2\frac{4i}{(1+2i)^2}

  1. Expand the denominator (1+2i)2(1+2i)^2.

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

  2. Rewrite the expression with the expanded denominator.

    4i−3+4i\frac{4i}{-3+4i}

  3. Multiply the numerator and denominator by the conjugate of the new denominator, which is −3−4i-3-4i.

    4i−3+4i×−3−4i−3−4i\frac{4i}{-3+4i} \times \frac{-3-4i}{-3-4i}

  4. Expand the numerator and the denominator.

    Numerator: 4i(−3−4i)=4i(−3)+4i(−4i)=−12i−16i2=−12i−16(−1)=16−12i4i(-3-4i) = 4i(-3) + 4i(-4i) = -12i - 16i^2 = -12i - 16(-1) = 16 - 12i
    Denominator: (−3+4i)(−3−4i)=(−3)2−(4i)2=9−16i2=9−16(−1)=9+16=25(-3+4i)(-3-4i) = (-3)^2 - (4i)^2 = 9 - 16i^2 = 9 - 16(-1) = 9 + 16 = 25

  5. Combine the simplified numerator and denominator.

    16−12i25=1625−1225i\frac{16-12i}{25} = \frac{16}{25} - \frac{12}{25}i

    The simplified form is \boxed{\text{\frac{16}{25} - \frac{12}{25}i}}.

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