Problem 84 - 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 \(0 .\) $$\frac{\sqrt{7}}{\sqrt{3}-\sqrt{7}}$$

Step-by-Step Solution

Answer

The expression rationalized is \(\frac{\sqrt{21} + 7}{-4}\).

Step 1: Identify the Problem

The given expression to simplify is \( \frac{\sqrt{7}}{\sqrt{3} - \sqrt{7}} \). The goal is to rationalize the denominator, meaning we need to eliminate the square roots from the denominator.

Step 2: Multiply by the Conjugate

To eliminate the square roots from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \(\sqrt{3} + \sqrt{7}\). This will utilize the difference of squares identity.

Step 3: Apply the Conjugate

Multiply the numerator and denominator: \( \frac{\sqrt{7}}{\sqrt{3} - \sqrt{7}} \times \frac{\sqrt{3} + \sqrt{7}}{\sqrt{3} + \sqrt{7}} = \frac{\sqrt{7} (\sqrt{3} + \sqrt{7})}{(\sqrt{3} - \sqrt{7})(\sqrt{3} + \sqrt{7})} \).

Step 4: Simplify the Numerator

The expression \(\sqrt{7} (\sqrt{3} + \sqrt{7})\) expands in the numerator to \(\sqrt{21} + 7\).

Step 5: Simplify the Denominator

In the denominator, use the difference of squares formula: \((\sqrt{3} - \sqrt{7})(\sqrt{3} + \sqrt{7}) = 3 - 7 = -4\).

Step 6: Combine and Simplify

Write the final expression: \(\frac{\sqrt{21} + 7}{-4}\). This is already simplified with the denominator rationalized.

Key Concepts

Radical Expressions

A radical expression is any mathematical expression that contains a radical symbol (√), which represents the square root or other roots like cube roots. When working with these expressions, it's essential to focus on simplification, especially when they are part of fractions.

Simplifying radical expressions often involves:

• Removing perfect squares from under the radical.
• Combining like radical terms.
• Rationalizing the denominator to make calculations easier.

In our original exercise, the radical expression is part of a fraction, with square roots in both the numerator and denominator. A common challenge is dealing with square roots in the denominator, which can complicate further operations or interpretations.

Difference of Squares

The difference of squares is a powerful algebraic identity used to simplify expressions in the form of \(a^2 - b^2\). The identity states:\[(a - b)(a + b) = a^2 - b^2\]This identity is particularly useful for rationalizing denominators in radical expressions. By multiplying by the conjugate, we can transform expressions that originally contained square roots into a straightforward subtraction of squares.

In our given solution, the denominator \((\sqrt{3} - \sqrt{7})\) is transformed into \((\sqrt{3})^2 - (\sqrt{7})^2\) using the difference of squares:\[3 - 7 = -4\]
By rearranging the radicals like this, the radicals in the denominator are effectively removed, simplifying the expression and making it easier to work with.

Conjugate in Algebra

In algebra, the conjugate of an expression refers to a specific modification, used to simplify expressions, particularly when dealing with complex or radical expressions. The conjugate of a binomial \(a - b\) is \(a + b\) and vice-versa.

The purpose of using a conjugate is to utilize the difference of squares identity, which allows us to eliminate radicals or imaginary numbers from expressions where they are not welcome, specifically in denominators.

In our exercise, the expression \(\frac{\sqrt{7}}{\sqrt{3} - \sqrt{7}}\) involved multiplying both the numerator and denominator by the conjugate of the denominator, \(\sqrt{3} + \sqrt{7}\). This helps to rationalize the denominator, resulting in an expression free from radicals in the denominator, transforming the fraction into \(\frac{\sqrt{21} + 7}{-4}\). This method ensures the expression is in a form that's easier to manage and interpret.