Problem 67 - Precalculus: Functions and Graphs

Question

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{6-3 i}{2+7 i} $$

Step-by-Step Solution

Answer

Expression: \(-\frac{9}{53} - \frac{48}{53}i\).

Step 1: Identify the Problem

We are given a complex fraction \( \frac{6-3i}{2+7i} \) and asked to express it in the form \( a + b i \), where \( a \) and \( b \) are real numbers. This requires separating the real and imaginary parts.

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2+7i \) is \( 2-7i \). So, we multiply: \( \frac{6-3i}{2+7i} \times \frac{2-7i}{2-7i} \).

Step 3: Calculate the Numerator

Calculate \( (6-3i)(2-7i) \). First distribute: \(6 \cdot 2 = 12\), \(6 \cdot (-7i) = -42i\), \(-3i \cdot 2 = -6i\), \(-3i \cdot (-7i) = 21i^2\). Remember that \(i^2 = -1\), so \(21i^2 = -21\). The numerator is \( 12 - 42i - 6i - 21 = -9 - 48i \).

Step 4: Calculate the Denominator

Execute \( (2+7i)(2-7i) \), which is a difference of squares: \( 2^2 - (7i)^2 = 4 - 49i^2 \). Since \( i^2 = -1 \), \(49i^2 = -49\), resulting in \( 4 + 49 = 53 \).

Step 5: Combine the Results

Now combine \( \frac{-9 - 48i}{53} \). Separate the real and imaginary parts: \( a = \frac{-9}{53} \) and \( b = \frac{-48}{53} \). Thus, the expression is \( -\frac{9}{53} - \frac{48}{53}i \).

Key Concepts

Complex Fraction

A complex fraction is similar to a regular fraction, but it involves complex numbers in either the numerator, the denominator, or both. In complex numbers, each component can be a combination of a real number and an imaginary number expressed in the general form \(a + bi\). To simplify a complex fraction, especially when the denominator includes an imaginary part, a common strategy is to multiply by the conjugate of the denominator. This helps in removing the imaginary component from the denominator, thereby simplifying the expression into a more standard form.

Real and Imaginary Parts

In the complex number format, \(a + bi\), \(a\) represents the real part, and \(b\) represents the imaginary part, with \(i\) being the imaginary unit where \(i^2 = -1\). To express a complex fraction like \(\frac{6-3i}{2+7i}\) in standard form, it is crucial to separate the real and imaginary components after simplification. The ultimate goal is to have the fraction appear as \(a + bi\), indicating the distinctly separated real and imaginary values. By simplifying the numerical and imaginary coefficients, you can clearly identify the values of \(a\) and \(b\). This makes the solution more intuitive.

Conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). This is a fundamental concept that helps in rationalizing complex fractions. When working with complex fractions, multiplying by the conjugate allows the imaginary parts to cancel out in the denominator due to the `difference of squares' property. Take the denominator \(2+7i\) from our original expression; its conjugate is \(2-7i\). Multiplying the numerator and the denominator of \(\frac{6-3i}{2+7i}\) by \(2-7i\) results in the elimination of the imaginary components from the denominator. This step brings the fraction closer to a fully simplified form.

Difference of Squares

The difference of squares is an algebraic identity used to simplify expressions of the form \((a+b)(a-b)\), resulting in \(a^2-b^2\). In the context of complex fractions, this identity becomes helpful when you multiply a complex number by its conjugate. For example, multiplying \((2+7i)(2-7i)\) utilizes the difference of squares: \((2)^2 - (7i)^2\). Calculating further: \(4 - 49i^2\). Since \(i^2 = -1\), it becomes \(4 + 49\). As a result, the expression's denominator is simplified to \(53\), a real number, thereby achieving the required rationalization.