Problem 26 - 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. $$ \sqrt{\frac{5 a}{18}} $$

Step-by-Step Solution

Answer

The simplest radical form is \( \frac{\sqrt{10a}}{6} \).

Step 1: Simplify the Fraction

First, we separate the square root of the numerator and the denominator in the expression: \( \sqrt{\frac{5a}{18}} \). This becomes \( \frac{\sqrt{5a}}{\sqrt{18}} \).

Step 2: Simplify the Denominator

Next, we need to simplify the denominator \( \sqrt{18} \). Note that \( 18 = 9 \times 2 \), and since \( \sqrt{9} = 3 \), we can simplify this to \( \sqrt{18} = 3\sqrt{2} \).

Step 3: Rationalize the Denominator

To simplify further, we need to rationalize the denominator. Multiply both the numerator and denominator by \( \sqrt{2} \) to eliminate the radical in the denominator: \[ \frac{\sqrt{5a} \times \sqrt{2}}{3\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{10a}}{6} \].

Step 4: Final Simplified Radical Form

Now \( \frac{\sqrt{10a}}{6} \) is in its simplest radical form. The numerator remains \( \sqrt{10a} \) as there are no perfect squares to extract, and the fraction is now simplified.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, it simply means making the fraction as simple as possible. A simplified fraction will have the smallest numbers in both the numerator and the denominator that still represent the same value. In terms of radical expressions, this process starts by evaluating and simplifying each part separately.

When you encounter a fraction under a square root, like \( \sqrt{\frac{5a}{18}} \), separate it into two separate square roots: one for the numerator and one for the denominator. This is done because simplifying each part separately might make it easier to see opportunities for further simplification:

• Numerator: \( \sqrt{5a} \)
• Denominator: \( \sqrt{18} \)

This breaks down the problem and lets you work on each part, ensuring the fraction is in its simplest form without hidden square roots lurking in the denominator. The goal here is to identify any factorable parts and ensure the fraction doesn't hide any unnecessary complexity.

Rationalizing the Denominator

Rationalizing the denominator involves removing radicals from the denominator of a fraction. A fractional expression is considered simplified when there are no radicals in the denominator.

To rationalize the denominator, look for a method to eliminate any radicals present. In our example, the denominator is \( 3\sqrt{2} \). To remove \( \sqrt{2} \), we multiply both the numerator and the denominator by \( \sqrt{2} \):

• This transforms the denominator from \( 3\sqrt{2} \) to \( 3 \times 2 = 6 \).
• The numerator will change accordingly, resulting in \( \sqrt{10a} \).

This method does not change the value of the expression, but it makes working with it much easier since conventional fractions are generally simpler to handle than fractions containing radicals. This process is crucial for reducing complexity and ensuring expressions are standardized for easier further calculations.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. Handling them often requires simplifying what’s inside the radical to its most basic components, particularly when variables are involved. This can make identifying any simplifications much clearer.

In our given problem \( \sqrt{\frac{5a}{18}} \), the expression initially looks dense. By simplifying what's inside, you're transforming the entire picture:

• \( \sqrt{5a} \) is as simple as it gets since neither \( 5 \) nor \( a \) are perfect squares.
• \( \sqrt{18} \) is simplified to \( 3 \sqrt{2} \), as \( 18 = 9 \times 2 \) and \( \sqrt{9} = 3 \).
• This simplification helps achieve final expressions, where the radicals are in their simplest form.

Engaging with radicals this way is about finding and extracting anything that can mitigate the radical’s complexity. It gives a cleaner and more useful form, ready for algebraic manipulation or just an easier read on your math tests or homework. Understanding these steps is essential for mastering radical expressions.