Problem 92 - Algebra and Trigonometry

Question

Rationalize the denominator. $$ \frac{1}{\sqrt{x}+1} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{x} - 1}{x - 1} \)

Step 1: Identify the Conjugate

To rationalize the denominator \( \sqrt{x} + 1 \), identify its conjugate. The conjugate of \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator of the expression \( \frac{1}{\sqrt{x}+1} \) by the conjugate, \( \sqrt{x} - 1 \). This results in: \[ \frac{1 \cdot (\sqrt{x} - 1)}{(\sqrt{x} + 1)(\sqrt{x} - 1)} \].

Step 3: Simplify the Denominator

The denominator \( (\sqrt{x} + 1)(\sqrt{x} - 1) \) simplifies using the difference of squares formula: \( a^2 - b^2 = (a-b)(a+b) \). This simplifies to \( x - 1 \).

Step 4: Simplify the Expression

The entire expression becomes: \[ \frac{\sqrt{x} - 1}{x - 1} \]. Now, the denominator no longer contains a radical.

Key Concepts

Conjugate

The concept of a conjugate is crucial in rationalizing denominators involving radicals. To find the conjugate of an expression like \( \sqrt{x} + 1 \), simply change the sign between the two terms. Thus, the conjugate is \( \sqrt{x} - 1 \). By multiplying by the conjugate, we take advantage of the algebraic properties that simplify the expression. This method helps eliminate radicals from the denominator, making the expression easier to work with.

Difference of Squares

The difference of squares is a powerful algebraic tool. It states that \( a^2 - b^2 = (a-b)(a+b) \). This formula applies when simplifying expressions involving conjugates. For the denominator \(( \sqrt{x} + 1)( \sqrt{x} - 1)\), apply the difference of squares:

• Let \( a = \sqrt{x} \) and \( b = 1 \).
• The expression becomes \( x - 1 \) as \( (\sqrt{x})^2 - 1^2 = x - 1 \).

Thus, the denominator is now free of radicals.

Simplification

Simplification is about reducing an expression to its simplest form. After rationalizing the denominator using the conjugate, the expression \( \frac{1}{\sqrt{x} + 1} \) transforms. By multiplying both the numerator and denominator by the conjugate \( \sqrt{x} - 1 \), the expression becomes \( \frac{\sqrt{x} - 1}{x - 1} \). The denominator is now a simple difference, \( x - 1 \), without any radicals. This "cleaned" version is easier to interpret and use in further mathematical processes.

Radicals

Radicals, like \( \sqrt{x} \), can be tricky in equations because they are not polynomials. Rationalizing the denominator involves eliminating radicals for simplicity. In the original problem, the radical is \( \sqrt{x} \) in the denominator.

• We "rationalize" by using its conjugate \( \sqrt{x} - 1 \), turning it into \( x - 1\) in the denominator.
• This avoids the complications radicals introduce in division and simplifies the structure of the expression.

Thus, rationalizing the denominator streamlines the expression into a more conventional form, free of radical terms.