Problem 59 - Intermediate Algebra

Question

Rationalize each denominator. $$ \frac{4}{\sqrt{6}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{2\sqrt{6}}{3} \).

Step 1: Identify the Need to Rationalize

We need to eliminate the square root from the denominator. The term \( \sqrt{6} \) is irrational, so we'll multiply both the numerator and the denominator by \( \sqrt{6} \) to rationalize it.

Step 2: Multiply by the Conjugate

Multiply the fraction \( \frac{4}{\sqrt{6}} \) by \( \frac{\sqrt{6}}{\sqrt{6}} \). This gives: \[\frac{4}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{4\sqrt{6}}{\sqrt{6} \times \sqrt{6}}.\]

Step 3: Simplify the Denominator

The denominator simplifies as \( \sqrt{6} \times \sqrt{6} = 6 \). Therefore, the expression becomes: \[\frac{4\sqrt{6}}{6}.\]

Step 4: Simplify the Fraction

The numerator and the denominator of the fraction \( \frac{4\sqrt{6}}{6} \) both have a common factor of 2. Simplifying gives: \[\frac{4\sqrt{6}}{6} = \frac{2\sqrt{6}}{3}.\]

Key Concepts

Understanding Irrational Numbers: What Makes them Special?

Irrational numbers are fascinating in the world of mathematics. They are numbers that cannot be expressed as simple fractions, meaning they cannot be written as a ratio of two integers.
For example, numbers such as \( \pi \) and \( \sqrt{2} \) are not easily broken down into neat fractional parts.

• These numbers often appear when dealing with square roots that aren't perfect squares.
• Their decimal expansions are non-terminating and non-repeating, which means the digits go on forever without forming a predictable pattern.
• In the context of the exercise, \( \sqrt{6} \) represents an irrational number because it does not have a whole number as a square root.

When you encounter an irrational number in the denominator, it is essential to rationalize it. This is a way to "tidy up" expressions by converting the denominator into a rational number.

Delving into Square Roots: The Basics

Square roots are one of the most fundamental operations in mathematics. A square root of a number \( x \) is a number \( y \) that, when multiplied by itself, gives \( x \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

• When dealing with non-perfect squares like 6, its square root is irrational.
• The square root operation is denoted by the symbol \( \sqrt{} \), which helps to identify numbers under the radical sign.
• In the exercise, \( \sqrt{6} \) cannot be simplified further into a precise integer, emphasizing its irrationality.

In problems requiring rationalization, square roots often appear in denominators. To remove them, you multiply by an expression that will yield a rational denominator.

Simplifying Fractions: A Key to Clarity

Simplifying fractions is about reducing them to their simplest form. This means making it impossible to divide the numerator and denominator by any common factor other than 1.

• It helps in making fractional expressions much easier to interpret and work with.
• For example, in the original exercise, after rationalizing the denominator, we ended up with \( \frac{4\sqrt{6}}{6} \). By identifying the common factor of 2, this fraction can be simplified to \( \frac{2\sqrt{6}}{3} \).
• Simplifying ensures that the fraction is more elegant and less complex.

The beauty of simplifying fractions is that it delivers results that are more comprehensible and manageable, especially in complex mathematical expressions.