Problem 78 - Algebra and Trigonometry

Question

\(77-82\) me Rationalize the denominator. $$ \frac{2}{x-\sqrt{5}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Denominator

The denominator in the expression \( \frac{2}{x-\sqrt{5}} \) is \( x-\sqrt{5} \). To rationalize this, we need to eliminate the square root from the denominator.

Step 2: Multiply by the Conjugate

To eliminate the square root, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( x-\sqrt{5} \) is \( x+\sqrt{5} \). Thus, the expression becomes \( \frac{2}{x-\sqrt{5}} \times \frac{x+\sqrt{5}}{x+\sqrt{5}} \).

Step 3: Simplify the Numerator

Multiply the numerator: \( 2 \times (x+\sqrt{5}) = 2x + 2\sqrt{5} \).

Step 4: Simplify the Denominator

Use the difference of squares formula \((a-b)(a+b) = a^2 - b^2\): \( (x-\sqrt{5})(x+\sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5 \).

Step 5: Combine and Simplify the Expression

Combine the simplified numerator and denominator. The rationalized expression is \( \frac{2x + 2\sqrt{5}}{x^2 - 5} \).

Key Concepts

Conjugate

In mathematics, the conjugate is a term used to simplify expressions that contain roots or complex numbers. Specifically, when rationalizing the denominator of a fraction, multiplying by the conjugate of the denominator can help eliminate radicals.
Think of the conjugate as a simple "flip" of the sign between two terms. For instance, if your denominator is of the form \( a-b \) (such as \( x-\sqrt{5} \)), the conjugate would be \( a+b \) (or \( x+\sqrt{5} \)).

• This works because multiplying conjugates utilizes the difference of squares formula, which helps remove the radical.
• The conjugate is particularly useful when you need to "clear" or "rationalize" a radical from the denominator.

Using conjugates is a handy technique in algebra and helps maintain the expression in a cleaner form or a form that is easier to interpret and use in further calculations.

Difference of Squares

The difference of squares is a handy algebraic identity. It states that the product of sums and differences of the same two terms equals the difference of their squares:
\[ (a-b)(a+b) = a^2 - b^2 \]
This formula quickly becomes your best friend when working with conjugates to rationalize denominators. Let's break it down:

• \( a \) is the first term, like \( x \) in our example.
• \( b \) is the second term, such as \( \sqrt{5} \).

When you multiply \( (x-\sqrt{5})(x+\sqrt{5}) \), you essentially use the difference of squares formula—a simple multiplication that transforms radicals into integers, here as \( x^2 - 5 \).
Using this can make your final expression less cluttered by eliminating radicals from the denominator.

Simplifying Expressions

Simplifying an expression aims to bring it to its simplest, most comprehensible form. After you multiply by the conjugate and apply the difference of squares, your expression may require some last cleaning up.
For the numerator, distribute any multipliers across terms. In the given example, you multiply \( 2 \) by both \( x \) and \( \sqrt{5} \) to get \( 2x + 2\sqrt{5} \).
The denominator becomes a straightforward integer expression, as seen in \( x^2 - 5 \).

• Examine the terms; double-check if they can be combined or re-simplified further.
• Ensure your final answer retains its accuracy without complicating the expression.

By achieving a neat and precise form, you prepare your work for any subsequent mathematical operations or evaluations needed. Simplifying also aids in enhancing comprehension and ensuring expressions are easily manageable.