(1/1-4i-2/1+i)(1-i/5+3i)

3 min read Jun 16, 2024

Simplifying Complex Expressions

This article will guide you through simplifying the complex expression:

(1/1-4i - 2/1+i)(1 - i/5 + 3i)

Step 1: Simplifying the Fractions

First, we need to simplify the fractions within the expression. We do this by multiplying the numerator and denominator of each fraction by the conjugate of the denominator.

For the first fraction (1/1-4i):
- The conjugate of 1-4i is 1+4i.
- (1/1-4i) * (1+4i)/(1+4i) = (1+4i)/(1 + 16) = (1+4i)/17
For the second fraction (2/1+i):
- The conjugate of 1+i is 1-i.
- (2/1+i) * (1-i)/(1-i) = (2-2i)/(1 + 1) = (2-2i)/2 = 1-i

Now our expression becomes:

(1+4i)/17 - (1-i)(1 - i/5 + 3i)

Step 2: Expanding the Expression

Next, we expand the expression by multiplying the terms:

(1+4i)/17 - (1 - i/5 + 3i - i + i^2/5 - 3i^2)

Step 3: Simplifying with i^2 = -1

Remember that i^2 = -1. We can substitute this into our expression:

(1+4i)/17 - (1 - i/5 + 3i - i - 1/5 + 3)

Step 4: Combining Like Terms

Now, we combine the real and imaginary terms:

((1 - 1 - 1/5 + 3) + (4 - 1/5 + 3 - 1)i)/17

Step 5: Final Simplification

Finally, we simplify the expression:

(17/5 + 29/5i)/17 = (17/85) + (29/85)i

Therefore, the simplified form of the expression (1/1-4i - 2/1+i)(1 - i/5 + 3i) is (17/85) + (29/85)i.