Problem 79 - Algebra and Trigonometry

Question

\(77-82\) me Rationalize the denominator. $$ \frac{2}{\sqrt{2}+\sqrt{7}} $$

Step-by-Step Solution

Answer

\(-\frac{2\sqrt{2}}{5} + \frac{2\sqrt{7}}{5}\) is the rationalized fraction.

Step 1: Identify the Conjugate

The rationalization of the denominator involves multiplying both the numerator and the denominator by 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 original fraction by \( \frac{\sqrt{2} - \sqrt{7}}{\sqrt{2} - \sqrt{7}} \): \[ \frac{2}{\sqrt{2} + \sqrt{7}} \cdot \frac{\sqrt{2} - \sqrt{7}}{\sqrt{2} - \sqrt{7}} = \frac{2(\sqrt{2} - \sqrt{7})}{(\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7})} \]

Step 3: Simplify the Denominator Using Difference of Squares

The denominator \( (\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7}) \) simplifies using the difference of squares: \[ (\sqrt{2})^2 - (\sqrt{7})^2 = 2 - 7 = -5 \]

Step 4: Distribute in the Numerator

Distribute 2 in the numerator: \[ 2\sqrt{2} - 2\sqrt{7} \]

Step 5: Write the Rationalized Fraction

Combine the results to write the rationalized fraction: \[ \frac{2\sqrt{2} - 2\sqrt{7}}{-5} \] which can be separated into: \[-\frac{2\sqrt{2}}{5} + \frac{2\sqrt{7}}{5} \]

Step 6: Simplify if Possible

Check if further simplification is possible. In this case, the expression is already in its simplified form.

Key Concepts

Conjugate

Rationalizing a denominator often requires using the conjugate of an expression. In mathematics, a conjugate refers to a pair where the two terms have the same components, except for the sign between them. For example, the conjugate of \[\sqrt{2} + \sqrt{7}\]is \[\sqrt{2} - \sqrt{7}.\]When we rationalize a denominator, it's crucial to understand the purpose of using the conjugate. By multiplying the numerator and the denominator by the conjugate, we leverage properties such as the difference of squares to eliminate the radicals. This process simplifies the expression's denominator, making it "rational," or free from radical components.

• Conjugates help to clear out irrational parts in denominators.
• Used to reduce radical terms in the final expression.

Using a conjugate simplifies expressions while preserving mathematical accuracy. This approach works because multiplying a pair of conjugates yields a neat numerical result, thanks to the magic of differences of squares.

Difference of Squares

The difference of squares is a fascinating algebraic concept that can simplify expressions involving radicals or complex numbers. This principle states that:\[(a + b)(a - b) = a^2 - b^2.\]In the original exercise, the denominator \[(\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7})\]becomes \[(\sqrt{2})^2 - (\sqrt{7})^2.\]Using difference of squares, square the individual terms:

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

Now, subtract the squares:\[2 - 7 = -5.\]Through understanding difference of squares, not only does the radical disappear in the denominator, but the structure becomes simpler and easier to manage. Reducing a complex expression using the difference of squares is a tool that highlights elegance in mathematical procedures.

Simplifying Expressions

Simplifying an expression involves combining like terms, reducing fractions, or eliminating radicals to make the expression easier to read and solve. This process includes rationalizing expressions like the one in the exercise, where the goal is to present the denominator in a simpler form.To begin with, multiply both the numerator and denominator by the conjugate of the denominator:\[\frac{2}{\sqrt{2} + \sqrt{7}} \cdot \frac{\sqrt{2} - \sqrt{7}}{\sqrt{2} - \sqrt{7}}\]This step incorporates both the conjugate and difference of squares principles. Next, distribute the number across terms:\[2(\sqrt{2} - \sqrt{7}) = 2\sqrt{2} - 2\sqrt{7}.\]After simplifying the denominator using the difference of squares, we have \[-5,\]making the expression:\[\frac{2\sqrt{2} - 2\sqrt{7}}{-5}.\]Breaking this fraction down further provides each term in its simplest format:

• \(-\frac{2\sqrt{2}}{5}\)
• \(+\frac{2\sqrt{7}}{5}\)

This form illustrates how multiplying by the conjugate and using algebraic rules like difference of squares can clean up complex radical expressions effectively.