Problem 65 - Algebra 2

Question

Simplify. $$ \frac{1-i \sqrt{2}}{1+i \sqrt{2}} $$

Step-by-Step Solution

Answer

\(-\frac{1}{3} - \frac{2}{3}i\sqrt{2}\)

Step 1: Identify the Conjugate

To simplify the expression \( \frac{1 - i\sqrt{2}}{1 + i\sqrt{2}} \), our goal is to eliminate the imaginary part from the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1 + i\sqrt{2} \) is \( 1 - i\sqrt{2} \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by \( 1 - i\sqrt{2} \). The expression becomes: \[ \frac{(1 - i\sqrt{2})(1 - i\sqrt{2})}{(1 + i\sqrt{2})(1 - i\sqrt{2})} \]

Step 3: Simplify the Denominator

Use the difference of squares formula \((a + b)(a - b) = a^2 - b^2\) to simplify the denominator: \[ (1 + i\sqrt{2})(1 - i\sqrt{2}) = 1^2 - (i\sqrt{2})^2 = 1 - (-2) = 1 + 2 = 3 \]

Step 4: Expand the Numerator

Expand the numerator: \[ (1 - i\sqrt{2})(1 - i\sqrt{2}) = 1^2 - 2 \cdot 1 \cdot i\sqrt{2} + (i\sqrt{2})^2 = 1 - 2i\sqrt{2} - 2 \] The expanded form simplifies to: \[ -1 - 2i\sqrt{2} \]

Step 5: Combine the Result

Combine the simplified numerator and denominator: \[ \frac{-1 - 2i\sqrt{2}}{3} \]

Step 6: Simplify the Final Expression

To simplify further, divide each term in the numerator by 3: \[ -\frac{1}{3} - \frac{2}{3}i\sqrt{2} \] This is the simplified form of the original expression.

Key Concepts

Conjugate in Complex Numbers

The conjugate of a complex number is a crucial concept in simplifying expressions involving complex numbers. A complex number is generally expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).
The conjugate of \( a + bi \) is \( a - bi \). It essentially involves changing the sign of the imaginary part while keeping the real part unchanged. This operation is helpful when we want to eliminate the imaginary part from the denominator of a complex fraction.
To illustrate, if we have \( 1 + i\sqrt{2} \), its conjugate is \( 1 - i\sqrt{2} \). By multiplying a complex number by its conjugate, we transform it into a real number because the cross terms cancel out, leaving only real components.
Using conjugates is a fundamental technique to simplify complex fractions and makes further computations easier to handle.

Difference of Squares

The difference of squares is a mathematical identity that simplifies the product of two binomials that are conjugates. The formula is \((a + b)(a - b) = a^2 - b^2\).
This identity is especially helpful when working with complex numbers and their conjugates. Applying the difference of squares allows us to simplify expressions such that the imaginary parts cancel out, leaving behind a real number.
In the example of \((1 + i\sqrt{2})(1 - i\sqrt{2})\), we can apply this identity as follows:

• Set \(a = 1\) and \(b = i\sqrt{2}\)
• Compute \(a^2 = 1^2 = 1\)
• Compute \(b^2 = (i\sqrt{2})^2 = -2\)
• Combine to get \(1 - (-2) = 1 + 2 = 3\)

This application shows how the difference of squares is used to simplify the denominator to a real number, making complex fractions much more manageable.

Simplifying Expressions

Simplifying complex expressions often involves combining concepts such as utilizing conjugates, applying identities like the difference of squares, and breaking down the expression into more manageable parts.
For complex fractions, the goal is often to remove the imaginary part from the denominator to simplify the expression's form. This simplification process involves several key steps:

• Identify the conjugate of the denominator.
• Multiply both the numerator and denominator by this conjugate to form real denominators.
• Apply algebraic identities to expand and simplify both the numerator and the denominator.
• Divide each term in the numerator by the resulting real number in the denominator.

Let's walk through the simplified process using the example \(\frac{1 - i\sqrt{2}}{1 + i\sqrt{2}}\):
1. Multiply by \(1 - i\sqrt{2}\), the conjugate of the denominator.
2. Apply the difference of squares to simplify the denominator.
3. Expand the resulting numerator. Use basic algebra to simplify it effectively.
4. Break down each term in the simplified numerator by the real result of the denominator.
Ultimately, simplifying these expressions enhances understanding and aids in working with complex numbers more accurately.