Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{1}{\sqrt{x-1}}$$

Step-by-Step Solution

Answer

\( \frac{\sqrt{x-1}}{x-1} \) has a rationalized denominator.

Step 1: Identify the Need for Rationalization

The given expression is \( \frac{1}{\sqrt{x-1}} \). Here, the denominator is \( \sqrt{x-1} \). The goal of rationalization is to remove the square root from the denominator.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{x-1} \). This will give \( \frac{1 \times \sqrt{x-1}}{\sqrt{x-1} \times \sqrt{x-1}} = \frac{\sqrt{x-1}}{x-1} \).

Step 3: Simplify the Expression

Since the denominator \( \sqrt{x-1} \times \sqrt{x-1} = x-1 \), the expression simplifies to \( \frac{\sqrt{x-1}}{x-1} \). Now, the denominator is rationalized because there is no square root in \( x-1 \).

Key Concepts

Denominators

In mathematics, a denominator is a crucial part of a fraction. It is the number or expression located below the fraction line (also known as the division bar) and serves as the divisor that indicates into how many parts the unit or whole is divided. In the expression \( \frac{1}{\sqrt{x-1}} \), \( \sqrt{x-1} \) is the denominator. Understanding the role of denominators is essential in operations on fractions like addition, subtraction, multiplication, and division. Rationalizing a denominator involves transforming it into a simpler or more "rational" form without a radical, such as a square root. This process is applied to make computations easier, especially when adding or subtracting fractions with similar denominators. By eliminating the square root, we make the denominator a real number, which simplifies further operations.

Square Root Removal

Removing a square root from the denominator, often called rationalization, is essential to simplify expressions. The problem with square roots in denominators is that they can complicate calculations, making them less intuitive. For example, if we have the expression \( \frac{1}{\sqrt{x-1}} \), the presence of \( \sqrt{x-1} \) can make it difficult to compare or compute operations with other fractions.To remove the square root, we need to manipulate the expression in a way that simplifies the root away. This typically involves multiplying the fraction by a version of 1 that includes the root (or its conjugate). For instance:

• Multiply both top and bottom by \( \sqrt{x-1} \).
• The outcome is that the square root in the denominator cancels out, leaving a rational number.

Such manipulation ensures that the expressional complexity is reduced without changing the value of the original fraction.

Conjugate Multiplication

Conjugate multiplication is a tool often used to rationalize expressions, especially when square roots are in play. A conjugate of a binomial like \( a + \sqrt{b} \) is \( a - \sqrt{b} \), and vice versa. Multiplying a binomial by its conjugate results in a difference of squares, a fundamental algebraic identity.In our problem \( \frac{1}{\sqrt{x-1}} \), although it's not a binomial, the technique of "conjugate" applies similarly since we need to eliminate the square root:

• By multiplying \( \frac{1}{\sqrt{x-1}} \) by \( \sqrt{x-1} \), we achieve a denominator of \( (\sqrt{x-1})^2 \).
• This simplifies to \( x-1 \), providing a rational denominator without a square root.

This method is valuable because it transforms expressions into more workable forms for further mathematical processing, such as integration or logarithmic differentiation.