Problem 24 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{3 x}{4 y}} $$

Step-by-Step Solution

Answer

The simplest radical form is \( \frac{\sqrt{3xy}}{2y} \).

Step 1: Separate the Inside of the Radical

Start by separating the radical into two parts, one for the numerator and one for the denominator.\[ \sqrt{\frac{3x}{4y}} = \frac{\sqrt{3x}}{\sqrt{4y}} \]

Step 2: Simplify the Denominator

Simplify the square root of the denominator. Since \(4y = 4 \times y\), and \(\sqrt{4} = 2\), extract this term from the radical:\[ \frac{\sqrt{3x}}{\sqrt{4y}} = \frac{\sqrt{3x}}{2\sqrt{y}} \]

Step 3: Rationalize the Denominator

To remove the square root from the denominator, multiply the numerator and the denominator by \(\sqrt{y}\):\[ \frac{\sqrt{3x}}{2\sqrt{y}} \times \frac{\sqrt{y}}{\sqrt{y}} = \frac{\sqrt{3xy}}{2y} \]

Step 4: Final Simplified Form

Combine all terms to express the radical in its simplest form. The expression is simplified and the radical is rationalized:\[ \sqrt{\frac{3x}{4y}} = \frac{\sqrt{3xy}}{2y} \]

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any radicals present in the denominator of a fraction. This is often necessary because having a radical in the denominator can make further calculations more complicated and less intuitive.

Here's how you can rationalize the denominator step by step:

• Identify the radical in the denominator: Look for any square roots or other radicals that are in the denominator of your fraction.
• Multiply by a form of 1: To eliminate the radical, multiply both the numerator and the denominator by the conjugate radical expression (essentially the radical itself in most cases).
• Simplify the expression: After multiplying, simplify both the numerator and the denominator to remove the radicals from the denominator. This often involves calculating the square of the radical which simplifies to a rational number.

This technique ensures that the denominator is free of radicals, making your fraction easier to work with in further mathematical operations.

Simplest Radical Form

The simplest radical form of an expression is a form where all the radicals are as simplified as possible, and no radicals remain in the denominator. This makes the expression easier to read and calculate.

To achieve this:

• Factor out perfect squares: Break down the number under the radical into its prime factors and look for perfect squares or cubes, depending on the index of the radical.
• Separate the radical: If the expression is a fraction, separate the numerator and denominator under their own radicals for simpler manipulation.
• Simplify each part: Take the square root of the perfect squares, and rewrite the expression so that radicals are as simplified as possible.

Finding the simplest radical form ensures that the expression is not only clean and tidy but also prepared for further algebraic manipulation or evaluation.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9.

Understanding square roots is crucial because they are foundational to the concept of radical expressions. Here's what you need to consider:

• Identifying perfect squares: Recognize numbers like 4, 9, 16, and their multiples, as they have whole number square roots. This can significantly simplify calculations.
• Using square roots in fractions: In cases where the radicand is a fraction, remember you can separate the square roots for both the numerator and the denominator.
• Simplifying further: Combine terms appropriately to minimize the expression, especially in equations where solving for a variable is needed.

This concept is at the heart of many algebraic processes and is an essential skill for solving and simplifying radical expressions efficiently.