Problem 27 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{2+\sqrt{2}}{3-\sqrt{2}}\)

Step-by-Step Solution

Answer

The rationalized and simplified form of the fraction is \( \frac{8 + 5\sqrt{2}}{7} \).

Step 1: Identify the Conjugate

The first step is to identify the conjugate of the denominator. The denominator is \(3 - \sqrt{2}\), so its conjugate will be \(3 + \sqrt{2}\).

Step 2: Multiply Numerator and Denominator by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate \(3 + \sqrt{2}\): \[ \frac{(2 + \sqrt{2})(3 + \sqrt{2})}{(3 - \sqrt{2})(3 + \sqrt{2})} \]

Step 3: Apply the Difference of Squares Formula

The denominator now becomes a difference of squares: \[ (3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \]. This simplifies the denominator to 7.

Step 4: Expand the Numerator

Expand the expression in the numerator: \[ (2 + \sqrt{2})(3 + \sqrt{2}) = 2 imes 3 + 2 imes \sqrt{2} + \sqrt{2} imes 3 + \sqrt{2} imes \sqrt{2} \]. Calculating each part, we get: \[ 6 + 2\sqrt{2} + 3\sqrt{2} + 2 = 8 + 5\sqrt{2} \].

Step 5: Combine and Simplify the Fraction

Combine your results from Steps 3 and 4 into a single fraction: \[ \frac{8 + 5\sqrt{2}}{7} \]. Since the numerator is in its simplest form, and the denominator is a prime number, this fraction is fully simplified.

Key Concepts

Understanding Conjugates

In mathematics, a conjugate is often used to simplify expressions that involve square roots. When you have a binomial expression like \(a + b\sqrt{c}\), its conjugate would be \(a - b\sqrt{c}\). The conjugate is vital in rationalizing denominators because it helps cancel out the square roots when used correctly.
For instance, when you multiply \((a + b\sqrt{c})(a - b\sqrt{c})\), you are essentially applying the conjugate of one of the terms to provide a rational expression. The process of doing this is crucial as it will eliminate the irrational parts in the denominator.

• Identify the expression you wish to rationalize.
• Find the conjugate of the denominator.
• Multiply both the numerator and denominator by the conjugate.

These steps will set you on a path to simplify expressions effectively.

The Power of Difference of Squares

The difference of squares is a mathematical property that states: \(a^2 - b^2 = (a + b)(a - b)\). This principle is exceptionally useful in the process of rationalizing denominators, especially when paired with conjugates.
Applying this property allows us to completely remove square roots from the denominator. In our example, multiplying the denominator \(3 - \sqrt{2}\) by its conjugate \(3 + \sqrt{2}\), results in:

• \((3)^2 - (\sqrt{2})^2 = 9 - 2 = 7\).

Here, the square roots have been effectively eliminated, leaving a rational number instead. Using the difference of squares is a strong tool in mathematics, transforming complex or irrational expressions into simpler forms.

Simplifying Fractions the Easy Way

Simplifying fractions is all about expressing the fraction in its simplest form. This means that the greatest common factor (GCF) of both the numerator and the denominator should be 1.
After rationalizing the denominator, which now is 7, our task is to check the numerator. Through expansion, the numerator simplifies to \(8 + 5\sqrt{2}\). Since the numerator and the denominator do not share any common factors, the fraction \(\frac{8 + 5\sqrt{2}}{7}\) is already simplified.

• Ensure both parts of the fraction have been completely reduced.
• Verify by checking for any common factors beside one.

Certainly, simplifying fractions is the last yet decisive step to ensure your expression is as clear and concise as possible.