Solve (3i-1)(1/3+i/2)

$$( 3 i - 1 ) ( \frac { 1 } { 3 } + \frac { i } { 2 } )$$

Evaluate

$$-\frac{11}{6}+\frac{1}{2}i\approx -1.833333333+0.5i$$

Solution Steps

Divide i by 2 to get \frac{1}{2}i.

$$\left(3i-1\right)\left(\frac{1}{3}+\frac{1}{2}i\right)$$

Use the distributive property to multiply 3i-1 by \frac{1}{3}+\frac{1}{2}i.

$$-\frac{3}{2}+i+\left(-\frac{1}{3}-\frac{1}{2}i\right)$$

Combine the real and imaginary parts in numbers -\frac{3}{2}+i and -\frac{1}{3}-\frac{1}{2}i.

$$-\frac{3}{2}-\frac{1}{3}+\left(1-\frac{1}{2}\right)i$$

Add -\frac{3}{2} to -\frac{1}{3}. Add 1 to -\frac{1}{2}.

$$-\frac{11}{6}+\frac{1}{2}i$$

Real Part

$$-\frac{11}{6} = -1\frac{5}{6} = -1.8333333333333333$$

Solution Steps

Divide i by 2 to get \frac{1}{2}i.

$$Re(\left(3i-1\right)\left(\frac{1}{3}+\frac{1}{2}i\right))$$

Use the distributive property to multiply 3i-1 by \frac{1}{3}+\frac{1}{2}i.

$$Re(-\frac{3}{2}+i+\left(-\frac{1}{3}-\frac{1}{2}i\right))$$

Combine the real and imaginary parts in numbers -\frac{3}{2}+i and -\frac{1}{3}-\frac{1}{2}i.

$$Re(-\frac{3}{2}-\frac{1}{3}+\left(1-\frac{1}{2}\right)i)$$

Add -\frac{3}{2} to -\frac{1}{3}. Add 1 to -\frac{1}{2}.

$$Re(-\frac{11}{6}+\frac{1}{2}i)$$

The real part of -\frac{11}{6}+\frac{1}{2}i is -\frac{11}{6}.

$$-\frac{11}{6}$$