Problem 95 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt{7}-\sqrt{2}}{\sqrt{2}+\sqrt{7}} $$

Step-by-Step Solution

Answer

The rationalized expression is \(-\frac{2\sqrt{14}}{5} + \frac{9}{5}\).

Step 1: Identify the expression to rationalize

We have the expression \( \frac{\sqrt{7}-\sqrt{2}}{\sqrt{2}+\sqrt{7}} \). Our goal is to rationalize the denominator.

Step 2: Determine the conjugate of the denominator

The denominator is \( \sqrt{2} + \sqrt{7} \). The conjugate of this is \( \sqrt{2} - \sqrt{7} \). Multiplying by the conjugate will help remove the radicals.

Step 3: Multiply the numerator and the denominator by the conjugate

Multiply both the numerator and the denominator by the conjugate, which is \( \sqrt{2} - \sqrt{7} \): \[\frac{(\sqrt{7}-\sqrt{2})(\sqrt{2}-\sqrt{7})}{(\sqrt{2}+\sqrt{7})(\sqrt{2}-\sqrt{7})}\]

Step 4: Simplify the numerator

Use the distributive property (also known as FOIL for binomials) to simplify the numerator: \[(\sqrt{7} - \sqrt{2})(\sqrt{2} - \sqrt{7}) = \sqrt{7}\sqrt{2} - \sqrt{7}\sqrt{7} - \sqrt{2}\sqrt{2} + \sqrt{2}\sqrt{7}\]This simplifies to: \[\sqrt{14} - 7 - 2 + \sqrt{14} = 2\sqrt{14} - 9\]

Step 5: Simplify the denominator

The denominator \((\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7})\) is a difference of squares, so it simplifies to: \[(\sqrt{2})^2 - (\sqrt{7})^2 = 2 - 7 = -5\]

Step 6: Write the final rationalized expression

The expression is now: \[\frac{2\sqrt{14} - 9}{-5}\]You can leave it in this form or distribute the negative sign. Thus: \[-\frac{2\sqrt{14}}{5} + \frac{9}{5}\]

Step 7: Verify that the denominator is rational

The denominator \(-5\) is a rational number, which confirms that the denominator has been successfully rationalized.

Key Concepts

Conjugate Multiplication

Conjugate multiplication is a valuable technique that is commonly used in rationalizing denominators, especially when dealing with expressions involving roots or radicals. The idea behind it is simple: to eliminate the radical expression in the denominator by multiplying it by its conjugate.

In mathematics, the conjugate of an expression is formed by changing the sign between two terms. For example, consider the expression with a denominator of \( \sqrt{2} + \sqrt{7} \). Its conjugate would be \( \sqrt{2} - \sqrt{7} \). By multiplying both the numerator and the denominator of the fraction by this conjugate, you can eliminate the radicals in the denominator.

This process leverages the fact that multiplying conjugates results in a difference of squares, ultimately simplifying the expression. This technique is not just limited to binomials with two terms, but it works excellently for any expression where the radicals need to be removed from the denominator.

Difference of Squares

The difference of squares is a mathematical identity that appears often when rationalizing denominators with conjugates. This identity states that the product of the sum and difference of the same two terms results in the difference of their squares. Mathematically, it's expressed as:
\[(a + b)(a - b) = a^2 - b^2\]
Applying this to the given exercise, let’s break it down:

• Our expression's denominator involves \( \sqrt{2} + \sqrt{7} \) and its conjugate, \( \sqrt{2} - \sqrt{7} \).
• When we multiply these two together, we get a difference of squares:
• \( (\sqrt{2})^2 - (\sqrt{7})^2 = 2 - 7 = -5 \)

This identity helps to eliminate the radicals in the denominator, leading to a simpler rational number, completing the rationalization process successfully. It’s a powerful tool when dealing with polynomial expressions and is one of the cornerstones of algebra.

Simplifying Radicals

Simplifying radicals is a crucial step in the process of rationalizing denominators. It involves simplifying the terms under the radical sign and ensures the expressions can be more easily worked with or compared.

In the context of this exercise, after applying the distributive property (i.e., FOIL method for binomials) to the numerator \((\sqrt{7} - \sqrt{2})(\sqrt{2} - \sqrt{7})\), we have an expression that initially includes terms like \(\sqrt{14} - 7 - 2 + \sqrt{14}\). Upon simplification, this results in:

• Combining like terms (such as the \(\sqrt{14}\) terms) and simplifying the integers.
• The result is \(2\sqrt{14} - 9\), which is an expression that has significantly simplified the radicals involved.

Such simplification helps not only rationalize the denominator but also provides a clear and simplified expression overall. Simplified radicals facilitate easier computation and understanding of the mathematical concepts involved.