Problem 103 - College Algebra with Modeling and Visualization

Question

Rationalize the denominator. $$ \frac{1}{3-\sqrt{2}} $$

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{3 + \sqrt{2}}{7}\).

Step 1: Identify the Conjugate

The first step in rationalizing the denominator is to identify the conjugate of the denominator. The original denominator is \(3 - \sqrt{2}\). The conjugate will have the same terms but with the opposite sign between them, so the conjugate is \(3 + \sqrt{2}\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator of the fraction by the conjugate we found. Essentially, this means multiplying the fraction by \(\frac{3 + \sqrt{2}}{3 + \sqrt{2}}\): \[ \frac{1}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} = \frac{3 + \sqrt{2}}{(3 - \sqrt{2})(3 + \sqrt{2})} \].

Step 3: Simplify the Denominator

Use the difference of squares formula, \((a-b)(a+b) = a^2 - b^2\), to simplify the denominator: \[ (3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \].

Step 4: Write the Final Rationalized Expression

After simplifying the denominator, write the final expression with the rationalized denominator: \[ \frac{3 + \sqrt{2}}{7} \]. This is the expression with the rationalized denominator.

Key Concepts

Conjugate

When working with square roots in denominators, especially in expressions like \(3 - \sqrt{2}\), the concept of a conjugate is very useful. A conjugate is formed by changing the sign between two terms. Here, the conjugate of \(3 - \sqrt{2}\) is \(3 + \sqrt{2}\). This change of sign makes it possible to eliminate the square root in the denominator upon multiplication.
Applying a conjugate is crucial when rationalizing denominators. It helps transform irrational or complex expressions into simpler ones. To rationalize, multiply both the numerator and the denominator of the fraction by this conjugate. Remember the golden rule in math: whatever you do to the denominator, you do to the numerator, ensuring the expression's value remains unchanged.

Difference of Squares

The difference of squares is a powerful formula used in algebra to simplify expressions. It states:

• \((a-b)(a+b) = a^2 - b^2\)

By multiplying the expression in the denominator \((3 - \sqrt{2})\) by its conjugate \((3 + \sqrt{2})\), we can apply this formula. Here, \(a\) is \(3\) and \(b\) is \(\sqrt{2}\). Upon substitution, the expression becomes \(3^2 - (\sqrt{2})^2\), simplifying to \(9 - 2 = 7\).The main advantage of using the difference of squares is that it eliminates the square roots from the denominator, making the expression easier to handle. It's a neat algebraic trick for simplifying complex fractions into something more manageable.

Simplifying Fractions

Simplifying fractions is a fundamental skill in mathematics that involves rewriting a fraction in its simplest form. In the context of rationalization, once the denominator is a simple whole number, it becomes straightforward to present the fraction in its most reduced form. For instance, after rationalizing the original fraction, the numerator becomes \(3 + \sqrt{2}\) and the denominator is \(7\). Since 7 is a prime number and does not divide into any element of the numerator, \(\frac{3 + \sqrt{2}}{7}\) is already simplified.
Always aim to simplify, as simplified fractions are not only easier to calculate with but also make the results clearer and more comprehensible. This process helps in maintaining the accuracy and simplicity of mathematical expressions.