Problem 86 - A Graphical Approach to College Algebra

Question

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{\sqrt{7}-1}{2 \sqrt{7}+4 \sqrt{2}}$$

Step-by-Step Solution

Answer

The rationalized expression is \(-\frac{7}{2} + \sqrt{14} + \frac{\sqrt{7}}{2} - \sqrt{2}\).

Step 1: Identify the Expression and the Goal

We need to rationalize the denominator of the expression \( \frac{\sqrt{7}-1}{2\sqrt{7}+4\sqrt{2}} \). Our goal is to eliminate the radicals in the denominator by multiplying both the numerator and the denominator by an appropriate conjugate.

Step 2: Determine the Conjugate

The conjugate of \(2\sqrt{7} + 4\sqrt{2}\) is \(2\sqrt{7} - 4\sqrt{2}\). We will use this conjugate to rationalize the denominator.

Step 3: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate \(2\sqrt{7} - 4\sqrt{2}\):\[\frac{(\sqrt{7} - 1)(2\sqrt{7} - 4\sqrt{2})}{(2\sqrt{7} + 4\sqrt{2})(2\sqrt{7} - 4\sqrt{2})}\]

Step 4: Simplify the Denominator

Use the difference of squares formula to simplify the denominator: \[(2\sqrt{7})^2 - (4\sqrt{2})^2 = 4 \cdot 7 - 16 \cdot 2 = 28 - 32 = -4\]The denominator simplifies to \(-4\).

Step 5: Simplify the Numerator

Distribute the terms in the numerator:\[(\sqrt{7})(2\sqrt{7}) - (\sqrt{7})(4\sqrt{2}) - 1(2\sqrt{7}) + 1(4\sqrt{2})\]Simplify each part:- \((\sqrt{7})(2\sqrt{7}) = 2 \times 7 = 14\)- \((\sqrt{7})(4\sqrt{2}) = 4\sqrt{14}\)- \(1(2\sqrt{7}) = 2\sqrt{7}\)- \(1(4\sqrt{2}) = 4\sqrt{2}\)Combine these to get: \[ 14 - 4\sqrt{14} - 2\sqrt{7} + 4\sqrt{2} \]

Step 6: Write the Final Rationalized Expression

Combine the simplified numerator and denominator: \[\frac{14 - 4\sqrt{14} - 2\sqrt{7} + 4\sqrt{2}}{-4}\]Divide each term in the numerator by \(-4\):\[-\frac{14}{4} + \frac{4\sqrt{14}}{4} + \frac{2\sqrt{7}}{4} - \frac{4\sqrt{2}}{4}\]Simplify further:\[-\frac{7}{2} + \sqrt{14} + \frac{\sqrt{7}}{2} - \sqrt{2}\]The final expression is:\[-\frac{7}{2} + \sqrt{14} + \frac{\sqrt{7}}{2} - \sqrt{2}\]

Step 7: Ensure Comprehensive Understanding

Look back at our steps to ensure that each operation was carried out correctly. Verify that the rationalized denominator is free of radicals and the expression is fully simplified.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is an essential technique in algebra that helps to eliminate radicals from the denominator of a fraction. Consider the expression \( \frac{\sqrt{7} - 1}{2\sqrt{7} + 4\sqrt{2}} \), our task is to make the denominator free of any radical expressions. This is achieved by multiplying both the numerator and denominator by what is known as the 'conjugate' of the denominator.

• Conjugate: The conjugate of a binomial like \( a\sqrt{x} + b\sqrt{y} \) is \( a\sqrt{x} - b\sqrt{y} \). This "flips" the sign between the terms while retaining the radical parts.
• The purpose of using a conjugate is to utilize the difference of squares formula, \( (a - b)(a + b) = a^2 - b^2 \), to eliminate radicals.

In our example, the conjugate of \( 2\sqrt{7} + 4\sqrt{2} \) is \( 2\sqrt{7} - 4\sqrt{2} \). When we multiply by this conjugate, the denominator simplifies to a whole number, thus eliminating the radicals. This process makes it easier to work with the expression, whether it's for solving equations or other algebraic operations.

Radical Expressions

Radical expressions involve numbers under a square root, cube root, or other roots, making them unique. They often appear in both the numerator and denominator of fractions. A radical expression like \( \sqrt{7} \) or \( 4\sqrt{2} \) contains both a coefficient and a radicand—the number inside the root symbol. Understanding the basic properties of radicals is vital:

• Product Rule: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This rule allows for the combination or separation of radical terms when multiplying.
• Quotient Rule: \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). You can divide under the same radical when simplifying.

When working with radical expressions in denominators, it's important to simplify them. Simplification might involve multiplying by a conjugate to eradicate the radicals. Knowing when to apply these rules enhances your algebraic manipulation skills, making more complex expressions easier to handle.

Simplification Techniques

Simplifying algebraic expressions is all about reducing terms to their simplest form. This not only makes the expressions easier to read but also more manageable in calculations. The original expression often gets altered by following specific simplification techniques.

• Distributive Property: Used when multiplying terms such as in the expression \((\sqrt{7} - 1)(2\sqrt{7} - 4\sqrt{2})\). Here, each term in the first bracket is multiplied by each term in the second bracket.
• Combine Like Terms: After distributing, you may end up with terms that can be combined, like \( 14 - 4\sqrt{14} - 2\sqrt{7} + 4\sqrt{2} \).
• Simplify Fraction Terms: Each term in the numerator is divided by the simplified value of the denominator. For terms like \(-\frac{7}{2}\), \(\sqrt{14}\), etc., make sure they are expressed in their simplest form.

Once these techniques are applied, the expression becomes more straightforward. Simplification often requires patience and attention to detail. However, the end result—a clearer and more concise expression—is well worth the effort.