Problem 59 - Elementary and 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 Irrational Denominator

The expression \( \frac{4}{\sqrt{6}} \) has an irrational denominator, which is \( \sqrt{6} \). Our goal is to rationalize this denominator.

Step 2: Multiply Numerator and Denominator by \( \sqrt{6} \)

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{6} \), i.e., \( \frac{4}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} \).

Step 3: Simplify the Expression

Upon multiplying, we get \( \frac{4\sqrt{6}}{6} \). This is because \( \sqrt{6} \times \sqrt{6} = 6 \).

Step 4: Reduce the Fraction

The last step is to simplify \( \frac{4\sqrt{6}}{6} \). Both the numerator (4) and the denominator (6) can be divided by 2. This results in \( \frac{2\sqrt{6}}{3} \).

Key Concepts

Exploring Irrational Numbers

Irrational numbers are a fascinating group of numbers that cannot be expressed as simple fractions or ratios of two integers. Unlike rational numbers, which can be neatly written as fractions like \( \frac{3}{4} \), irrational numbers go on forever in a non-repeating pattern. Some well-known examples of irrational numbers include \( \pi \), usually approximated to 3.14159, and the square root of any non-perfect square, such as \( \sqrt{2} \) or \( \sqrt{6} \).

Here are some interesting points about irrational numbers:

• Irrational numbers are non-repeating and non-terminating when written as decimals.
• They cannot be exactly represented as fractions of integers.
• They can be often found in geometry, such as the diagonal of a square, which is \( \sqrt{2} \).

In the given exercise, \( \sqrt{6} \) is an irrational number found in the denominator. To "rationalize" it, we transform it so it no longer has an irrational number in the denominator.

Understanding Square Roots

Square roots are a key mathematical concept, involving finding a number that, when multiplied by itself, results in the original number. For example, \( \sqrt{25} \) is 5 because \( 5 \times 5 = 25 \). Square roots can be rational or irrational depending on the number involved. If the number is a perfect square, such as 4, 9, or 16, its square root is rational. However, if the number is not a perfect square, like 2, 3, or 6, the square root becomes irrational.

Consider these points about square roots:

• Square roots of perfect squares are integers.
• Square roots of non-perfect squares are irrational.
• Operations on square roots follow specific algebraic rules, like \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \).

In our problem, \( \sqrt{6} \) appears in the denominator and it’s an irrational number, making computations less straightforward. By using multiplication with the same square root, \( \sqrt{6} \cdot \sqrt{6} = 6 \), we eliminate the irrationality in the denominator, moving towards a simplified expression.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest form where the numerator and the denominator are at their smallest co-prime values. This process can sometimes involve removing common factors shared between the numerator and the denominator. By simplifying a fraction, calculations become easier, and expressions are neater.

Here are the basic steps to simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• Ensure that the resulting fraction cannot be reduced further.

In the exercise provided, after rationalizing the denominator, the expression \( \frac{4\sqrt{6}}{6} \) simplifies further by finding that both 4 (numerator) and 6 (denominator) share a common factor of 2. By dividing both terms by 2, we obtain the simplified form \( \frac{2\sqrt{6}}{3} \), which is the final result.