How to Simplify Complex Fractions

Oct 13, 2025 by miodeya.com 115 views

**Simplify: $\frac{3+1i}{-2-5i}$** =====================================================

What is the Simplification of Complex Fractions?

In mathematics, complex fractions are a type of expression that involves division of two complex numbers. Simplifying complex fractions is an essential skill in algebra and calculus, as it allows us to express complex expressions in a simpler form. In this article, we will simplify the complex fraction $\frac{3+1i}{-2-5i}$ .

Why is Simplifying Complex Fractions Important?

Simplifying complex fractions is important because it helps us to:

Express complex expressions in a simpler form
Perform calculations more easily
Understand the behavior of complex functions
Solve equations and inequalities involving complex numbers

How to Simplify Complex Fractions

To simplify a complex fraction, we can use the following steps:

Multiply the numerator and denominator by the conjugate of the denominator
Simplify the resulting expression
Express the result in the form $a+bi$

Simplifying the Complex Fraction

Let's simplify the complex fraction $\frac{3+1i}{-2-5i}$ .

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

The conjugate of the denominator $-2-5i$ is $-2+5i$ . We multiply the numerator and denominator by the conjugate of the denominator:

\frac{3+1i}{-2-5i} \cdot \frac{-2+5i}{-2+5i}

Step 2: Simplify the Resulting Expression

We simplify the resulting expression:

\frac{(3+1i)(-2+5i)}{(-2-5i)(-2+5i)}

=\frac{-6+15i-2i+5i^2}{4+10i-10i-25i^2}

=\frac{-6+13i-5}{4+25}

=\frac{-11+13i}{29}

Step 3: Express the Result in the Form $a+bi$

We express the result in the form $a+bi$ :

\frac{-11+13i}{29} = -\frac{11}{29} + \frac{13}{29}i

Conclusion

In this article, we simplified the complex fraction $\frac{3+1i}{-2-5i}$ using the steps outlined above. We multiplied the numerator and denominator by the conjugate of the denominator, simplified the resulting expression, and expressed the result in the form $a+bi$ . Simplifying complex fractions is an essential skill in algebra and calculus, and we hope that this article has provided a clear and concise explanation of the process.

Frequently Asked Questions

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a+bi$ is $a-bi$ .

Q: Why do we multiply the numerator and denominator by the conjugate of the denominator?

A: We multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part of the denominator.

Q: How do we simplify a complex fraction?

A: We simplify a complex fraction by multiplying the numerator and denominator by the conjugate of the denominator, simplifying the resulting expression, and expressing the result in the form $a+bi$ .

Q: What is the importance of simplifying complex fractions?

A: Simplifying complex fractions is important because it helps us to express complex expressions in a simpler form, perform calculations more easily, understand the behavior of complex functions, and solve equations and inequalities involving complex numbers.

Q: Can we simplify complex fractions using other methods?

A: Yes, we can simplify complex fractions using other methods, such as using the polar form of complex numbers or using the exponential form of complex numbers. However, the method outlined above is the most common and straightforward method.