Problem 83 - Algebra and Trigonometry

Question

\(83-88=\) Rationalize the denominator. $$ \begin{array}{llll}{\text { (a) } \frac{1}{\sqrt{6}}} & {\text { (b) } \frac{3}{\sqrt{2}}} & {} & {\text { (c) } \frac{9}{\sqrt{3}}}\end{array} $$

Step-by-Step Solution

Answer

(a) \( \frac{\sqrt{6}}{6} \), (b) \( \frac{3\sqrt{2}}{2} \), (c) \( 3\sqrt{3} \).

Step 1: Understand the Problem

We need to rationalize the denominator for each of the given fractions: \( \frac{1}{\sqrt{6}} \), \( \frac{3}{\sqrt{2}} \), and \( \frac{9}{\sqrt{3}} \). Rationalizing involves removing the square root from the denominator by multiplying by an appropriate form of 1.

Step 2: Rationalize Option (a)

For \( \frac{1}{\sqrt{6}} \), multiply both the numerator and the denominator by \( \sqrt{6} \) to rationalize the denominator:\[\frac{1}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6}.\]

Step 3: Rationalize Option (b)

For \( \frac{3}{\sqrt{2}} \), multiply both the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator:\[\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}.\]

Step 4: Rationalize Option (c)

For \( \frac{9}{\sqrt{3}} \), multiply both the numerator and the denominator by \( \sqrt{3} \) to rationalize the denominator:\[\frac{9}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}.\]

Step 5: Simplify Results

The simplified and rationalized forms of the fractions are:(a) \( \frac{\sqrt{6}}{6} \)(b) \( \frac{3\sqrt{2}}{2} \)(c) \( 3\sqrt{3} \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. In mathematics, a fraction is composed of two numbers: a numerator and a denominator. The numerator is the top number, representing the number of parts considered, whereas the denominator is the bottom number, indicating how many parts the whole is divided into.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator, meaning that it represents 3 parts out of a total of 4.
It's crucial when working with fractions to understand operations such as addition, subtraction, multiplication, and division. When performing these operations, you often need a common denominator, especially in addition and subtraction.

• **Simplifying fractions:** this involves reducing the fraction to its smallest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• **Complex fractions:** involve a fraction in either the numerator, the denominator, or both. They often require simplification before further operations.
• **Equivalent fractions:** are different fractions that represent the same value, typically found by multiplying or dividing the numerator and denominator by the same number.

Without a solid understanding of fractions, tasks such as rationalizing the denominator become unnecessarily complicated.

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \). The square root of a number \( x \) is expressed as \( \sqrt{x} \). Understanding square roots is essential in various mathematical operations, including simplifying expressions and solving equations.
When dealing with square roots in fractional expressions, especially in the denominator, you often want to "rationalize" the fraction. This means removing the square root from the denominator to make calculations easier and the expression more understandable.
The process involves:

• **Multiplying by a form of 1:** This means multiplying the fraction by \( \frac{\sqrt{a}}{\sqrt{a}} \), where \( \sqrt{a} \) is the square root in the denominator.
• **Eliminating the square root:** Upon multiplication, the square root is squared, leaving a whole number in the denominator.

For instance, in \( \frac{1}{\sqrt{6}} \), multiplying by \( \frac{\sqrt{6}}{\sqrt{6}} \) results in \( \frac{\sqrt{6}}{6} \). Thus, understanding square roots enables effective simplification of complex mathematical expressions.

Simplifying Expressions

Simplifying expressions is the process of rewriting them in a more concise and comprehensive form, allowing for easier interpretation and solving. This is vital in mathematics to reduce complexity and enhance clarity. Simplifying fractions, radical expressions, or any mathematical problem follows similar fundamental processes.
When simplifying expressions with square roots, especially in fractions, follow these steps:

• **Identify like terms:** Before simplifying, ensure that you recognize parts of the expression that can be combined.
• **Rationalize the denominator:** As discussed, this involves removing any square roots in the denominator.
• **Combine and reduce:** If possible, add or subtract like terms, simplifying the fraction to its lowest terms.

Let’s take \( \frac{9}{\sqrt{3}} \) as an example. By multiplying by \( \frac{\sqrt{3}}{\sqrt{3}} \) to rationalize, you get \( \frac{9\sqrt{3}}{3} \). Further simplifying gives \( 3\sqrt{3} \). This shows how simplification turns a more complex expression into an easier form, enhancing understanding and utility in mathematical applications.