Simplifying Complex Fractions: (62i)/(2+4i)
This article will guide you through the process of simplifying the complex fraction (62i)/(2+4i).
Understanding Complex Numbers
Complex numbers are numbers 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.
Simplifying Complex Fractions
To simplify a complex fraction, we need to eliminate the imaginary term from the denominator. This is achieved by multiplying both the numerator and denominator by the complex conjugate of the denominator.
The Complex Conjugate
The complex conjugate of a complex number a + bi is a  bi.
Applying the Steps

Identify the denominator's complex conjugate: The complex conjugate of (2 + 4i) is (2  4i).

Multiply both the numerator and denominator by the complex conjugate:
(6  2i) / (2 + 4i) * (2  4i) / (2  4i)

Expand the numerator and denominator:
(12  24i  4i + 8i²) / (4  8i + 8i  16i²)

Remember that i² = 1:
(12  28i  8) / (4 + 16)

Simplify the expression:
(4  28i) / 20

Express the result in the standard form (a + bi):
(1/5)  (7/5)i
Conclusion
The simplified form of the complex fraction (62i)/(2+4i) is (1/5)  (7/5)i. This process demonstrates how to work with complex numbers and simplify expressions involving them.