Math

QuestionCalculate the result of the complex division: (38+135i)÷(1110i)(-38+135 i) \div(11-10 i).

Studdy Solution

STEP 1

Assumptions1. We are given a complex number division problem (38+135i)÷(1110i)(-38+135 i) \div(11-10 i). We will use the formula for dividing complex numbers \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^+d^}

STEP 2

First, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number (c+di)(c+di) is (cdi)(c-di).
38+135i1110i×11+10i11+10i\frac{-38+135i}{11-10i} \times \frac{11+10i}{11+10i}

STEP 3

Now, distribute the numerator and simplify.
(38+135i)(11+10i)=38113810i+135i11+135i10i(-38+135i)(11+10i) = -38 \cdot11 -38 \cdot10i +135i \cdot11 +135i \cdot10i

STEP 4

implify the expression.
=418380i+148i+1350i2= -418 -380i +148i +1350i^2

STEP 5

Remember that i2=1i^2 = -1, so substitute 1-1 for i2i^2 in the equation.
=418380i+1485i1350= -418 -380i +1485i -1350

STEP 6

Combine like terms.
=1768+1105i= -1768 +1105i

STEP 7

Now, distribute the denominator and simplify.
(1110i)(11+10i)=1111+1110i10i1110i10i(11-10i)(11+10i) =11 \cdot11 +11 \cdot10i -10i \cdot11 -10i \cdot10i

STEP 8

implify the expression.
=121+110i110i100i2=121 +110i -110i -100i^2

STEP 9

Again, remember that i2=i^2 = -, so substitute - for i2i^2 in the equation.
=121100=121 -100

STEP 10

Combine like terms.
=221=221

STEP 11

Now, divide the numerator by the denominator to get the final answer.
1768+1105i221=8+5i\frac{-1768 +1105i}{221} = -8 +5iSo, (38+135i)÷(1110i)=8+5i(-38+135 i) \div(11-10 i) = -8 +5i.

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