Problem 13 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Rationalized expression is \( \frac{\sqrt{x}}{x} \).

Step 1: Understand the Problem

The problem requires us to rationalize the denominator of the expression \( \frac{1}{\sqrt{x}} \). Rationalizing the denominator means transforming the expression so that there are no radicals (square roots) in the denominator.

Step 2: Multiply by the Conjugate

To get rid of the square root in the denominator, multiply both the numerator and the denominator by \( \sqrt{x} \), which is the same as the square root in the denominator. This will help eliminate the square root from the denominator.

Step 3: Perform the Multiplication

Carry out the multiplication: \[\frac{1}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{1 \times \sqrt{x}}{\sqrt{x} \times \sqrt{x}}\]This gives us \( \frac{\sqrt{x}}{x} \). The square root in the denominator is canceled since \( \sqrt{x} \times \sqrt{x} = x \).

Step 4: Simplify the Expression

The expression \( \frac{\sqrt{x}}{x} \) is already in its simplified form, with no radicals in the denominator.

Key Concepts

Radicals

A radical is a symbol that is used to represent the root of a number. The most common radical is the square root, represented by the symbol \( \sqrt{} \). Radicals help us find values that, when multiplied by themselves a certain number of times, give us the original number. For example, \( \sqrt{16} \) means finding a number that, when squared, equals 16; in this case, it's 4.

Radicals are very useful in mathematics but can sometimes complicate expressions, especially in denominators. Rationalizing involves eliminating radicals from the denominators to simplify expressions. This makes them easier to work with in further calculations or algebraic manipulations.

Denominators

Every fraction has a numerator and a denominator, where the denominator is the bottom part of the fraction. It tells us into how many parts the whole is divided. For example, in the fraction \( \frac{1}{2} \), 2 is the denominator, indicating two equal parts.

In some mathematical expressions, having a denominator with a radical can make calculations complex. For example, \( \frac{1}{\sqrt{x}} \) has a denominator of \( \sqrt{x} \). To rationalize such expressions, multiply both the numerator and the denominator by the correspondent radical. This process removes the radical from the denominator, making the expression easier to work with.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. It’s often represented using the radical symbol \( \sqrt{} \). For instance, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

Understanding square roots is essential for rationalizing denominators that contain radicals. For example, in rationalizing \( \frac{1}{\sqrt{x}} \), the objective is to multiply by \( \sqrt{x} \) to achieve an integer in the denominator. This conversion transforms the denominator from \( \sqrt{x} \times \sqrt{x} = x \), eliminating the square root. Thus, rationalizing expressions simplifies calculations and ensures that the resulting fractions are easier to interpret and use in math problems.