Problem 104 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ \frac{-4 i}{2-6 i} $$

Step-by-Step Solution

Answer

The result is \( \frac{3}{5} - \frac{1}{5}i \) in the form \( a+bi \).

Step 1: Identify the Problem

We need to simplify the complex fraction \( \frac{-4i}{2-6i} \) and express the result in the form \( a+bi \), where \( a \) and \( b \) are real numbers.

Step 2: Find the Conjugate

The denominator is \( 2-6i \). The conjugate of this complex number is \( 2+6i \). Multiplying by the conjugate will help us eliminate the imaginary part from the denominator.

Step 3: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator:\[ \frac{-4i}{2-6i} \times \frac{2+6i}{2+6i} = \frac{-4i(2+6i)}{(2-6i)(2+6i)} \].

Step 4: Simplify the Denominator

Use the formula \( (a-bi)(a+bi) = a^2 + b^2 \) to simplify:\[ (2-6i)(2+6i) = 2^2 + (6)^2 = 4 + 36 = 40 \].

Step 5: Expand the Numerator

Distribute \(-4i\) in the numerator:\[ -4i(2+6i) = (-4i \times 2) + (-4i \times 6i) = -8i + 24 \].

Step 6: Combine the Result

Now combine both parts:\[ \frac{-8i + 24}{40} = \frac{-8i}{40} + \frac{24}{40} = \frac{3}{5} - \frac{1}{5}i \].

Step 7: Present the Solution in the Required Form

The simplified form of the expression in \( a+bi \) is:\[ \frac{3}{5} - \frac{1}{5}i \]

Key Concepts

complex conjugate

The complex conjugate is a crucial concept in dealing with complex numbers. For any complex number, the complex conjugate is formed by changing the sign between its real and imaginary components.

For example, if we have a complex number expressed as \(a + bi\), its complex conjugate will be \(a - bi\). Notice how only the imaginary part changes sign.

In the exercise, we dealt with a denominator of \(2 - 6i\). Its conjugate is \(2 + 6i\). By multiplying the numerator and the denominator by this conjugate, we remove the imaginary parts from the denominator.

• This results in a real number in the denominator.
• It simplifies calculation by turning complex divisions into simpler fractions.
• It preserves the equality of fractions: we multiply by \(1\) in the form of a complex number over its conjugate.

Applying complex conjugates allows us to express any complex fraction in a simplified form, crucial for further simplifying or solving equations.

complex fraction

Complex fractions involve numerators and/or denominators that contain complex numbers.

To simplify a complex fraction, such as \(\frac{-4i}{2-6i}\), we aim to remove the imaginary unit from the denominator via multiplication by its conjugate. This not only makes calculations more straightforward but also aligns results with standard mathematical form.

When we multiply both the numerator and denominator by the conjugate of the denominator, we convert the denominator into a real number.

• This process involves distributing and simplifying.
• The final expression is often easier to interpret or further use.
• Mathematically transforming fractions to remove complex numbers in denominators is essential in complex analysis.

The result ensures that we express the division of complex numbers clearly, preserving its nature but in a more benefitted form for further operations.

simplification of complex expressions

Simplifying complex expressions involves breaking down expressions where complex numbers are used into their simplest form. The goal is to express the results in the standard form, \(a + bi\).

Starting with a complex expression like \(\frac{-4i}{2-6i}\), simplification steps include using the complex conjugate and proper distribution. This transforms the expression into \(\frac{3}{5} - \frac{1}{5}i\).

Key steps include:

• Finding conjugates to eliminate imaginary numbers in denominators.
• Distributing the imaginary unit appropriately across terms.
• Combining and simplifying results into the form of \(a+bi\), where \(a\) and \(b\) are real numbers.

With such techniques, any complex expression can be simplified, easing understanding and further calculation.