# The expression 3 + 2 i 1 − 4 i in the form of a + b i .

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter I, Problem 8E
To determine

## To evaluate: The expression 3+2i1−4i in the form of a+bi.

Expert Solution

The value of the expression 3+2i14i is 517+1417i.

### Explanation of Solution

Obtain the complex conjugate of number 14i.

Use the property of conjugates z+w¯=z¯+w¯ and zw¯=z¯w¯ to simplify the expression as follows.

14i¯=1¯4i¯=1¯(4¯(i¯))=1+4i[i¯=i]

Thus, the complex conjugate of the number 14i is 1+4i.

Multiply numerator and denominator of the expression 3+2i14i by the complex conjugate of the denominator 14i and simplify the expression as follows.

3+2i14i=3+2i14i(1+4i1+4i)=3+12i+2i+(4i)(2i)12(4i)2=3+14i+8i2116i2=3+14i+8(1)116(1)[i2=1]

Further simplified as,

3+2i14i=3+14i81+16=5+14i17=517+1417i

Thus, the value of the expression 3+2i14i is 517+1417i.

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