Problem 80 - Algebra and Trigonometry

Question

\(77-82\) me Rationalize the denominator. $$ \frac{1}{\sqrt{x}+1} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Need for Rationalization

The given expression is \( \frac{1}{\sqrt{x}+1} \). The denominator contains a square root term, \( \sqrt{x} \), which is irrational. To rationalize, we need to eliminate the square root from the denominator.

Step 2: Find the Conjugate of the Denominator

The conjugate of an expression like \( a + b \) is \( a - b \). Here, the denominator is \( \sqrt{x} + 1 \), so the conjugate is \( \sqrt{x} - 1 \).

Step 3: Multiply Numerator and Denominator by the Conjugate

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

Step 4: Simplify the Denominator

Apply the difference of squares formula to the denominator: \((\sqrt{x})^2 - 1^2 = x - 1\). Now the expression becomes:\[\frac{\sqrt{x} - 1}{x - 1}\]

Step 5: Simplify the Expression

Since the expression \( \frac{\sqrt{x} - 1}{x - 1} \) is already fully simplified after rationalization, no further simplification is required.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In the given problem, we have a square root in the denominator, specifically \( \sqrt{x} \). Square roots can create some challenges in mathematical expressions, especially when they appear in the denominator, because they are irrational numbers. This means that square roots often continue indefinitely without repeating, making accurate calculation difficult. By rationalizing the denominator, we convert this irrational number into a simpler form.

Conjugate

A conjugate is a key tool in rationalizing denominators containing square roots. For any binomial expression like \( a + b \), the conjugate is \( a - b \). Conjugates are used to eliminate the square root from the denominator of fractions. In our given expression, the conjugate of the denominator \( \sqrt{x} + 1 \) is \( \sqrt{x} - 1 \). By multiplying both the numerator and the denominator by the conjugate, we can effectively remove the square root, paving the way for simplification. This approach works because it leverages the properties of conjugates effectively.

Difference of Squares

The difference of squares is a powerful algebraic identity that applies to expressions of the form \( a^2 - b^2 \). This identity tells us that \( a^2 - b^2 = (a + b)(a - b) \). In our problem, after multiplying by the conjugate, we used this identity on the term \( (\sqrt{x} + 1)(\sqrt{x} - 1) \). This results in \( (\sqrt{x})^2 - 1^2 \), simplifying it to \( x - 1 \). The difference of squares makes the expression much easier to handle, as it removes the irrational component from the denominator.

Simplify Expression

Simplifying an expression is the process of making it as straightforward as possible. After using the difference of squares identity, the expression becomes \( \frac{\sqrt{x} - 1}{x - 1} \). At this point, the expression is fully simplified because the denominator no longer contains a square root, and the fraction is in its simplest form. Simplification helps in clearer interpretation and evaluation of the expression, and often makes further calculations more manageable. Always look to simplify expressions after each operation for the most understandable result.