Problem 91 - Algebra and Trigonometry

Question

Rationalize the denominator. $$ \frac{2}{\sqrt{2}+\sqrt{7}} $$

Step-by-Step Solution

Answer

The rationalized expression is \(-\frac{2\sqrt{2}}{5} + \frac{2\sqrt{7}}{5}\).

Step 1: Identify the Conjugate

The given expression is \( \frac{2}{\sqrt{2} + \sqrt{7}} \). To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \( \sqrt{2} + \sqrt{7} \) is \( \sqrt{2} - \sqrt{7} \).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply the numerator and the denominator of the expression by the conjugate \( \sqrt{2} - \sqrt{7} \): \[\frac{2}{\sqrt{2} + \sqrt{7}} \times \frac{\sqrt{2} - \sqrt{7}}{\sqrt{2} - \sqrt{7}} = \frac{2(\sqrt{2} - \sqrt{7})}{(\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7})}\]

Step 3: Simplify the Denominator

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to simplify the denominator:\[(\sqrt{2})^2 - (\sqrt{7})^2 = 2 - 7 = -5\]

Step 4: Expand and Simplify the Numerator

Expand the numerator:\[2(\sqrt{2} - \sqrt{7}) = 2\sqrt{2} - 2\sqrt{7}\]

Step 5: Write the Final Expression

Combine the simplified numerator and denominator:\[\frac{2\sqrt{2} - 2\sqrt{7}}{-5} = -\frac{2\sqrt{2}}{5} + \frac{2\sqrt{7}}{5}\]This is the final simplified expression with a rationalized denominator.

Key Concepts

Conjugate in Algebra

When we talk about the conjugate in algebra, we refer to flipping the sign of the terms within a binomial expression. This is often used to help simplify expressions, particularly when dealing with irrational numbers in denominators. For instance, if you have an expression like \( \sqrt{2} + \sqrt{7} \), its conjugate would be \( \sqrt{2} - \sqrt{7} \).

The role of the conjugate is crucial when rationalizing denominators. By multiplying the numerator and the denominator of a fraction by the conjugate of the denominator, it eliminates the irrational part in the denominator, making it a rational number. This works because the product of a number and its conjugate results in a difference of squares, which simplifies the expression greatly.

Difference of Squares

The difference of squares is a powerful algebraic tool often employed when rationalizing denominators. This formula works on the principle that the product of two binomials with opposite signs results in the difference of two squares. Mathematically, it's expressed as:

• \((a+b)(a-b) = a^2 - b^2\)

When you have an expression like \( (\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7}) \), the difference of squares rule states that it will simplify to:

• \((\sqrt{2})^2 - (\sqrt{7})^2 = 2 - 7 = -5\)

The beauty of this process is that it turns an otherwise complicated root expression into a simple integer or rational number, which is much easier to work with.

Simplifying Radical Expressions

In algebra, simplifying radical expressions is an essential skill that helps to streamline mathematical expressions for easier manipulation and comprehension. Simplifying radicals often involves removing irrational numbers from the denominator by using techniques like rationalizing.

This process can include

• Multiplying by conjugates to eliminate radicals from the denominator.
• Simplifying square roots by recognizing perfect squares within the radicand (the number inside the root).
• Combining like terms, if possible, once the radicals are simplified.

In the given exercise, we start by simplifying the denominator using its conjugate, which leads to a simple integer. Then, we simplify the numerator by distributing and combining terms, resulting in a clean fractional expression: \(-\frac{2\sqrt{2}}{5} + \frac{2\sqrt{7}}{5}\). Each step breaks down the complexity of the original scenario into manageable parts, ultimately making the equation easier to interpret and solve.