Problem 121 Simplify. $$\frac{6}{1+\frac{3... [FREE SOLUTION] (2024)

Expert verified

Complex numbers are numbers that have two parts: a real part and an imaginary part. The imaginary unit is denoted by \(i\), where \(i = \sr{-1}\). For example, in the expression \(2 + 3i\), the number 2 is the real part, and \(3i\) is the imaginary part.
Key points about complex numbers:

• The real part is a real number.
• The imaginary part is a real number multiplied by \(i\).
• Complex numbers follow the same arithmetic rules as real numbers, but including \(i\). For instance, \(i^2 = -1\).

Understanding how to manipulate complex numbers is essential when simplifying expressions that include them. In this exercise, we deal with a complex fraction that requires careful manipulation of the imaginary part.

Rationalizing the denominator means eliminating the imaginary part from the denominator of a fraction. This is done by multiplying the numerator and denominator by the conjugate of the denominator.
Steps to rationalize the denominator:

• Identify the complex conjugate: for any complex number \(a + bi\), its conjugate is \(a - bi\).
• Multiply both the numerator and the denominator by the conjugate.
• Use the formula for the difference of squares: \((a + bi)(a - bi) = a^2 - b^2i^2 = a^2 + b^2\) since \(i^2 = -1\).

For example, in our exercise, the denominator is \(1 - 3i\). Its conjugate is \(1 + 3i\), so we multiply both the numerator and the denominator by \(1 + 3i\) to eliminate the imaginary part from the denominator.

Fractions in mathematics represent parts of a whole. Simplifying fractions means making them as simple as possible. In our exercise, we simplify a complex fraction:
\text{1} The outer fraction is \(\frac{6}{1 + \frac{3}{i}}\). To simplify, follow these steps:

• First, simplify any fractions within the numerator or the denominator.
• Next, rationalize the denominator if it includes complex numbers.
• After that, combine the results if possible.

Breaking down and understanding each part of a fraction helps in gradually simplifying complex expressions, just as you see in the exercise provided.
Result: By following these steps, we reduce \(\frac{6}{1 + \frac{3}{i}}\) to \(\frac{3}{5} + \frac{9i}{5}\).