Problem 115 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ \frac{\sqrt{5}-i \sqrt{3}}{\sqrt{5}+i \sqrt{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 1 - \frac{\sqrt{15}}{4}i \).

Step 1: Identify the Need for Rationalizing the Denominator

The expression involves a complex number in the denominator, \( \sqrt{5} + i \sqrt{3} \). To simplify, we need to get rid of the imaginary part in the denominator by rationalizing it.

Step 2: Multiply by the Conjugate of the Denominator

To eliminate the imaginary part from the denominator, multiply the numerator and denominator by the conjugate of the denominator, \( \sqrt{5} - i \sqrt{3} \).\[\frac{\sqrt{5} - i \sqrt{3}}{\sqrt{5} + i \sqrt{3}} \times \frac{\sqrt{5} - i \sqrt{3}}{\sqrt{5} - i \sqrt{3}} \]

Step 3: Simplify the Numerator

In the numerator, use the distributive property (FOIL method) to simplify:\[ (\sqrt{5} - i \sqrt{3})(\sqrt{5} - i \sqrt{3}) = (\sqrt{5})^2 - 2i\sqrt{5}\sqrt{3} + (i\sqrt{3})^2 \]Simplify further:\[= 5 - 2i\sqrt{15} - 3i^2 \]Since \( i^2 = -1 \), this becomes:\[= 5 - 2i\sqrt{15} + 3 \]Thus, the simplified numerator is \( 8 - 2i\sqrt{15} \).

Step 4: Simplify the Denominator

The denominator is the product of the complex number and its conjugate, which simplifies to a real number:\[(\sqrt{5} + i \sqrt{3})(\sqrt{5} - i \sqrt{3}) = (\sqrt{5})^2 - (i \sqrt{3})^2 \]Expanding, we get:\[= 5 - (-3) \]Simplify further:\[= 5 + 3 = 8\]Thus, the denominator is 8.

Step 5: Combine and Simplify

Now divide the simplified numerator by the simplified denominator:\[\frac{8 - 2i\sqrt{15}}{8} = 1 - \frac{2\sqrt{15}}{8}i = 1 - \frac{\sqrt{15}}{4}i \]The final answer, in the form \( a + bi \), is \( 1 - \frac{\sqrt{15}}{4}i \).

Key Concepts

Rationalizing Denominators

Rationalizing the denominator is a technique used to eliminate complex numbers from the denominator of a fraction. It's especially useful for simplifying expressions to make them easier to work with or more understandable.
To rationalize the denominator, follow these steps:

• Identify the complex number in the denominator. A complex number has the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
• Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\).
• When you multiply a complex number by its conjugate, the imaginary components cancel out, leaving only a real number in the denominator. This process eliminates any imaginary parts from the denominator, simplifying the expression.

In our example, the expression \(\frac{\sqrt{5} - i \sqrt{3}}{\sqrt{5} + i \sqrt{3}}\) was rationalized by multiplying by \(\sqrt{5} - i \sqrt{3}\), the conjugate of the denominator, resulting in a simpler and more workable form.

Imaginary Numbers

Imaginary numbers are crucial to understanding complex numbers. The basic element of imaginary numbers is the imaginary unit \(i\), which is defined by the equation \(i^2 = -1\). This definition allows us to handle calculations involving the square roots of negative numbers.
In mathematics, especially when dealing with quadratic equations that do not have real solutions, imaginary numbers provide a way to express these solutions. In addition:

• An imaginary number can be represented as \(bi\), where \(b\) is a real number.
• When combined with a real number, imaginary numbers form complex numbers, written in the form \(a + bi\).

In our original exercise, the denominator of \(\sqrt{5} + i\sqrt{3}\) was a complex number with an imaginary component \(i\sqrt{3}\). By rationalizing, we simplified its form, which maintained the overall mathematical properties while making the expression easier to manage.

Conjugate of Complex Numbers

The conjugate of a complex number is an integral concept when working with complex numbers, especially in rationalizing denominators. Given a complex number in the form \(a + bi\), its conjugate is \(a - bi\).
Conjugates are useful because:

• Multiplying a complex number by its conjugate results in a real number. This product is calculated as \((a + bi)(a - bi) = a^2 + b^2\).
• This property is what helps to eliminate the imaginary part when rationalizing denominators of complex fractions.
• The conjugate also has applications in determining magnitude or modulus of complex numbers since \(|a + bi|^2 = (a + bi)(a - bi) = a^2 + b^2\).

In the given problem, the conjugate \(\sqrt{5} - i\sqrt{3}\) was crucial in simplifying the expression \(\frac{\sqrt{5} - i \sqrt{3}}{\sqrt{5} + i \sqrt{3}}\) by converting the denominator into a real number, thus facilitating easier calculations.