Simplifying Complex Fractions: (7+3i)/(2+6i)
This article will guide you through the process of simplifying the complex fraction (7+3i)/(2+6i).
Understanding Complex Numbers
Before we dive into the simplification, let's refresh our understanding of complex numbers. A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of 1.
Rationalizing the Denominator
To simplify the complex fraction, we need to eliminate the imaginary term from the denominator. This process is known as rationalizing the denominator. We achieve this by multiplying both the numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of (2+6i) is (26i).
The Calculation

Multiply numerator and denominator by the conjugate:
[(7+3i)/(2+6i)] * [(26i)/(26i)]

Expand using the distributive property (FOIL method):
[(14  42i  6i + 18i²)/ (4  12i + 12i  36i²)]

Simplify by substituting i² = 1:
[(14  48i  18)/(4 + 36)]

Combine real and imaginary terms:
[(32  48i)/40]

Simplify by dividing both numerator and denominator by their greatest common factor (8):
(4  6i)/5
Result
Therefore, the simplified form of (7+3i)/(2+6i) is (4  6i)/5. This representation is in the standard form of a complex number, a + bi, where a = 4/5 and b = 6/5.