Problem 3 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{1}{\sqrt{3}}\)

Step-by-Step Solution

Answer

The rationalized fraction is \( \frac{\sqrt{3}}{3} \).

Step 1: Identify the Need to Rationalize

Recognize that the fraction \( \frac{1}{\sqrt{3}} \) has an irrational number in the denominator, \( \sqrt{3} \). Our goal is to make this denominator rational by rationalizing it.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \), the radical term that appears in the denominator: \[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \] This step is essentially multiplying the fraction by 1, which doesn't change its value, but modifies its form.

Step 3: Multiply the Numerator and Denominator

Carry out the multiplication separately for the numerator and the denominator. - The numerator becomes \( 1 \times \sqrt{3} = \sqrt{3} \).- The denominator becomes \( \sqrt{3} \times \sqrt{3} = 3 \). Thus, we arrive at the intermediate fraction: \[ \frac{\sqrt{3}}{3} \]

Step 4: Confirm Simplest Form

Check that \( \frac{\sqrt{3}}{3} \) is in its simplest form, as it has no further reducible terms and the denominator is rational.

Key Concepts

Simplifying Fractions

Simplifying fractions is like cleaning up a messy room; everything should be in its simplest and most straightforward form. Fractions show a division between two numbers. A fraction is simplified when the top number (numerator) and the bottom number (denominator) are as small as possible yet maintain the same value or meaning of the fraction.
To simplify fractions:

• Determine the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

It's essential to simplify fractions to ensure they are easy to interpret and work with in calculations. For instance, \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \) because both 4 and 8 can be divided by 4, their GCD. When simplifying radicals like \( \frac{\sqrt{3}}{3} \), check if any further reduction is possible. Since \( \sqrt{3} \) is in its simplest form, so is \( \frac{\sqrt{3}}{3} \).

Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction because their decimal expansions neither terminate nor repeat. These numbers often arise from roots, like the square root of numbers that aren't perfect squares.
For example, \(\sqrt{2}\) and \(\sqrt{3}\) are irrational since they can't be written as exact fractions. Working with them involves special techniques, especially when they appear in fractions.When dealing with fractions like \(\frac{1}{\sqrt{3}}\), the presence of an irrational number in the denominator prompts the need to rationalize it. Irrational numbers are precise in nature. Comprehending them involves understanding they are non-repeating, non-terminating decimals, making them unique compared to their rational counterparts.

Radical Expressions

Radical expressions include the square root or higher roots of numbers. They can often be cumbersome, especially when used in equations or fractions. Simplifying radical expressions often involves making the expression easier to work with.

• "Simplifying" might mean reducing the radicand, the number under the root, or rationalizing the denominator of a fraction.
• Rationalizing involves removing radicals from the denominator by multiplying both numerator and denominator by a suitable form of 1, like \( \sqrt{3}/\sqrt{3} \), clarifying the expression.

Using the example \(\frac{1}{\sqrt{3}}\), when we rationalize, we turn it into \(\frac{\sqrt{3}}{3}\), where the root is only in the numerator. This method helps manage equations and make calculations clearer. Simplifying radical expressions ensures the math looks cleaner and is easier to follow.