91Ó°ÊÓ

Simplifying fractions is an essential skill in mathematics, including when dealing with complex numbers. In this example, we have the fraction \(\frac{2}{3-2i}\). To simplify, we need to manipulate the fraction so that we no longer have a complex number in the denominator. This is done through a series of steps involving multiplication, expansion, and simplification. By multiplying the numerator and the denominator by the conjugate of the denominator, the complex part in the denominator is eliminated. Simplifying the numerator and the denominator separately allows us to ultimately split the fraction into real and imaginary parts. This process ensures the fraction is in the simplest form, making it easier to interpret and work with.

In the case of complex numbers, the conjugate is crucial for simplification. The conjugate of a complex number \(a+bi\) is \(a-bi\). In our exercise, the denominator \(3-2i\) involves a complex number. To eliminate the imaginary part, multiply both the numerator and the denominator by the conjugate \(3+2i\). This process results in:

• Numerator: \(2 \times (3 + 2i) = 6 + 4i\)
• Denominator: \((3-2i)(3+2i) = 3^2 - (2i)^2 = 9 - 4(-1) = 13\)

The result is a simpler fraction \(\frac{6+4i}{13}\). Conjugation helps eliminate the complex part in the denominator, turning it into a real number, which greatly aids in simplification.

A critical formula in algebra, especially when dealing with complex numbers, is the difference of squares. It states that \((a-b)(a+b) = a^2 - b^2\). This formula is used to simplify the denominator in our example. When multiplying \(3-2i\) by its conjugate \(3+2i\), we apply the difference of squares formula:

• First term: \(3^2 = 9\)
• Second term: \((2i)^2 = 4i^2 = 4(-1) = -4\)
• Applying the formula: \((3-2i)(3+2i) = 9 + 4 = 13\)

uThe result, 13, is a real number simplifying our denominator. Using the difference of squares in such scenarios helps eliminate imaginary numbers in the denominator, streamlining the fraction into a more manageable form.