(6-2i)/(2+4i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (6-2i)/(2+4i)

This article will guide you through the process of simplifying the complex fraction (6-2i)/(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

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

2. Multiply both the numerator and denominator by the complex conjugate:

   (6 - 2i) / (2 + 4i) * (2 - 4i) / (2 - 4i)

3. Expand the numerator and denominator:

   (12 - 24i - 4i + 8i²) / (4 - 8i + 8i - 16i²)

4. Remember that i² = -1:

   (12 - 28i - 8) / (4 + 16)

5. Simplify the expression:

   (4 - 28i) / 20

6. Express the result in the standard form (a + bi):

   (1/5) - (7/5)i

Conclusion

The simplified form of the complex fraction (6-2i)/(2+4i) is (1/5) - (7/5)i. This process demonstrates how to work with complex numbers and simplify expressions involving them.