(10+4i)-(3-2i)/(6-7i)(1-2i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions

This article will guide you through the process of simplifying the complex expression:

(10 + 4i) - (3 - 2i) / (6 - 7i)(1 - 2i)

Step 1: Simplifying the denominator

We begin by simplifying the denominator, which involves multiplying two complex numbers. To do this, we use the distributive property (FOIL method):

(6 - 7i)(1 - 2i) = (6 * 1) + (6 * -2i) + (-7i * 1) + (-7i * -2i)

Simplifying further:

= 6 - 12i - 7i + 14i²

Recall that i² = -1. Substituting this in:

= 6 - 12i - 7i - 14 = -8 - 19i

Step 2: Simplifying the fraction

Now we have:

(10 + 4i) - (3 - 2i) / (-8 - 19i)

To divide complex numbers, we multiply both numerator and denominator by the complex conjugate of the denominator. The complex conjugate of (-8 - 19i) is (-8 + 19i).

= (10 + 4i) - [(3 - 2i) * (-8 + 19i)] / [(-8 - 19i) * (-8 + 19i)]

Expanding the numerator and denominator:

= (10 + 4i) - [(-24 + 38i + 16i - 38i²)] / [64 - 152i + 152i - 361i²]

= (10 + 4i) - [-24 + 54i + 38] / [64 + 361]

= (10 + 4i) - [14 + 54i] / 425

Step 3: Simplifying the entire expression

Now we can simplify the entire expression:

= (10 + 4i) - (14/425 + 54i/425)

= (10 - 14/425) + (4 - 54/425)i

= (4210/425 - 14/425) + (1680/425 - 54/425)i

= 4196/425 + 1626/425i

Therefore, the simplified form of the expression (10 + 4i) - (3 - 2i) / (6 - 7i)(1 - 2i) is 4196/425 + 1626/425i.