(2-7i)/(3-4i)

Jasonbradley

2 min read

·

Jun 16, 2024

13

Share

Simplifying Complex Fractions: (2-7i)/(3-4i)

This article will guide you through the process of simplifying the complex fraction (2-7i)/(3-4i).

Understanding Complex Numbers

Before diving into the simplification, let's understand the concept of 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 multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a - bi.

Steps:

1. Identify the complex conjugate: The complex conjugate of (3-4i) is (3+4i).
2. Multiply numerator and denominator by the conjugate: (2-7i)/(3-4i) * (3+4i)/(3+4i)
3. Expand using the distributive property (FOIL): (6 + 8i - 21i - 28i²)/(9 + 12i - 12i - 16i²)
4. Simplify using i² = -1: (6 - 13i + 28)/(9 + 16)
5. Combine real and imaginary terms: (34 - 13i)/25
6. Express in standard form (a + bi): (34/25) - (13/25)i

Therefore, the simplified form of (2-7i)/(3-4i) is (34/25) - (13/25)i.

Conclusion

Simplifying complex fractions involves multiplying both the numerator and denominator by the complex conjugate of the denominator. This process eliminates the imaginary component from the denominator, resulting in a simpler form of the complex number.