Problem 31 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{26+39 i}{2-3 i} $$

Step-by-Step Solution

Answer

The result is \(-5 + 12i\).

Step 1: Identify the Objective

The goal is to simplify the given expression and write it in the form \(a + b i\). To do this, we will use the process of rationalizing the denominator.

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - 3i\) is \(2 + 3i\). We multiply as follows:\[\frac{26+39i}{2-3i} \times \frac{2+3i}{2+3i}\]

Step 3: Expand the Numerator

Expand \((26 + 39i)(2 + 3i)\) by using the distributive property:\[26 \times 2 + 26 \times 3i + 39i \times 2 + 39i \times 3i\]Simplify:\[52 + 78i + 78i + 117i^2\]Since \(i^2 = -1\), this becomes:\[52 + 156i - 117\] which simplifies to \[-65 + 156i\].

Step 4: Calculate the Denominator

Expand \((2 - 3i)(2 + 3i)\) using the formula \((a-b)(a+b) = a^2 - b^2\):\[(2)^2 - (3i)^2 \]This is equivalent to:\[ 4 - 9(-1) = 4 + 9 = 13 \].

Step 5: Divide Real and Imaginary Parts

The expression can now be rewritten as:\[\frac{-65 + 156i}{13} = \frac{-65}{13} + \frac{156i}{13}\]Simplify each part:\[-5 + 12i\].

Step 6: Write in Form \(a+bi\)

Thus, the simplified expression in the form \(a+bi\) is:\[-5 + 12i\].

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is an essential process in algebra, especially useful when dealing with complex numbers. It makes calculations simpler and results cleaner by removing complex numbers from the denominator. We achieve this by multiplying both the numerator and denominator by the conjugate of the denominator.
This method takes advantage of the identity \( (a-b)(a+b) = a^2-b^2 \). When applied to complex numbers, this identity helps eliminate the imaginary part from the denominator, turning it into a real number. This step greatly simplifies the division process.

• First, identify the conjugate of the denominator.
• Multiply both the numerator and the denominator by this conjugate.
• Expand both parts using distributive property or FOIL (First, Outside, Inside, Last) method.

By rationalizing the denominator in our problem, the expression \( \frac{26+39i}{2-3i} \) becomes much simpler to handle.

Complex Conjugate

The complex conjugate is a critical concept when working with complex numbers. For a complex number \(a + bi\), its complex conjugate is \(a - bi\).
This simple transformation involves changing the sign of the imaginary part. The complex conjugate is fundamental in simplifying expressions, particularly when rationalizing denominators. When a complex number is multiplied by its conjugate, the result is a real number, which simplifies division.

• The conjugate of \(2 - 3i\) is \(2 + 3i\).
• Multiplying a number by its conjugate effectively uses the difference of squares identity.
• This ensures that any imaginary components cancel out.

This concept enables clearer interpretations and manipulations of complex expressions, bringing them into more manageable forms.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the cornerstone of complex numbers. It is defined by the property that \(i^2 = -1\).
This definition allows us to extend the set of real numbers to include solutions to equations that would otherwise have no real solutions, such as \(x^2 + 1 = 0\). When dealing with expressions involving complex numbers, knowing how \(i\) behaves is vital.
In computations involving complex numbers:

• Recognize that \(i^2 = -1\) to simplify expressions.
• When multiplying complex numbers, remember that terms involving \(i^2\) should be replaced with \(-1\).
• This substitution transforms and simplifies terms, aiding in reaching the standard form \(a+bi\).

Understanding \(i\) is fundamental for working through complex arithmetic and is integral in arriving at accurate solutions.