Problem 79 - Intermediate Algebra

Question

Rationalize each numerator. See Example 7. $$ \frac{\sqrt{2}-1}{\sqrt{2}+1} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{1}{3 + 2\sqrt{2}} \).

Step 1: Identify the Conjugate of the Numerator

To rationalize the numerator, identify the conjugate of the numerator \(\sqrt{2} - 1\). The conjugate is \(\sqrt{2} + 1\).

Step 2: Multiply by the Conjugate of the Numerator

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

Step 3: Simplify the Numerator Using Difference of Squares

Use the difference of squares formula \((a - b)(a + b) = a^2 - b^2\) to simplify the numerator. Here, \(a = \sqrt{2}\) and \(b = 1\), so\[(\sqrt{2})^2 - 1^2 = 2 - 1 = 1\]. The numerator simplifies to 1.

Step 4: Expand the Denominator

Multiply out the denominator \((\sqrt{2} + 1)(\sqrt{2} + 1)\), which is the square of a binomial. This can be expanded as\[(\sqrt{2})^2 + 2\cdot\sqrt{2}\cdot 1 + 1^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2}\].

Step 5: Write the Final Rationalized Expression

Substitute the simplified numerator and expanded denominator back into the fraction to get the rationalized form. The expression is\[\frac{1}{3 + 2\sqrt{2}}\].

Key Concepts

Conjugate Pairs

In mathematics, conjugate pairs are expressions that only differ by the sign between two terms. This concept is incredibly useful in simplifying expressions, particularly when dealing with roots and complex numbers. For example, if we have the expression \(a + b\), its conjugate would be \(a - b\). The importance of conjugates lies in their ability to help eliminate square roots or other irrational numbers in denominators or numerators.

• The conjugate of \( \sqrt{2} - 1 \) is \( \sqrt{2} + 1 \)
• Multiplying an expression by its conjugate often makes it simpler
• Through this process, irrational parts can often be removed

By multiplying the numerator by its conjugate, you create an expression that can often be simplified using another powerful algebraic concept: the difference of squares.

Difference of Squares

The difference of squares is an algebraic identity expressed by the formula \(a^2 - b^2 = (a-b)(a+b)\). This formula is crucial when multiplying conjugates because it simplifies the expression by removing the square roots. When you see an expression like \((\sqrt{2} - 1)(\sqrt{2} + 1)\), you can apply this rule directly.

• Here, \(a^2 = (\sqrt{2})^2 = 2\) and \(b^2 = 1^2 = 1\)
• The formula gives us \(2 - 1 = 1\)
• This simplification helps in getting rid of the square roots from the rationalizing process

Understanding how to apply the difference of squares can make working with polynomials and simplifying expressions much easier, allowing you to rationalize an expression swiftly.

Simplifying Expressions

Simplifying expressions is the process of reducing an expression to its simplest form. This often involves combining like terms, canceling factors, and using algebraic identities such as the ones we've discussed. In our exercise, after applying the difference of squares to simplify the numerator to 1, the next step is to tackle the denominator.

• The denominator \( (\sqrt{2} + 1)(\sqrt{2} + 1) \) is expanded to \(2 + 2\sqrt{2} + 1\)
• This combines to \(3 + 2\sqrt{2}\)
• Now our expression becomes \(\frac{1}{3 + 2\sqrt{2}}\)

By simplifying both the numerator and the denominator, you achieve a rationalized expression that is much easier to interpret and work with in subsequent calculations. Mastering these steps will enhance your ability to handle complex algebraic expressions more confidently.