Problem 39 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{19}{4}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{19}}{2}\)

Step 1: Identify the components inside the square root

We start with the expression \(\sqrt{\frac{19}{4}}\). This is a square root of a fraction, where the numerator is 19 and the denominator is 4.

Step 2: Separate the square root of the fraction

Recall the property of radicals that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Apply it here to separate the square root into two parts: \(\sqrt{19}\) and \(\sqrt{4}\).

Step 3: Simplify each radical separately

Calculate the square roots: \(\sqrt{4} = 2\) because 4 is a perfect square. \(\sqrt{19}\) cannot be simplified further as 19 is a prime number.

Step 4: Write in simplest radical form

Combine the results from Step 3: \(\frac{\sqrt{19}}{2}\). No further simplification is possible, so this is the final simplest radical form.

Key Concepts

Square Root Concept

The square root is a fundamental concept in mathematics. It refers to a value that, when multiplied by itself, results in the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. When dealing with square roots, you are often looking for a number that can evenly multiply to give a perfect square. To understand square roots better, here are a few helpful points:

• The symbol used for square root is \(\sqrt{}\).
• Perfect squares are numbers like 1, 4, 9, 16, and so on, whose square roots are whole numbers.
• For non-perfect squares, the square root will result in an irrational number, meaning it cannot be expressed as a simple fraction.

In solving problems, it's crucial to identify whether the number inside a square root is a perfect square, as this will help determine how to proceed with simplification.

Simplifying Radicals

Simplifying radicals is the process of transforming a complicated square root expression into a simpler or more manageable one. This is often necessary for efficient calculation and understanding.To simplify radicals:

• Firstly, determine if the number is a perfect square. If it is, like 4 or 16, it can be completely simplified into an integer (2 for 4, 4 for 16).
• If the number isn’t a perfect square, like 19, you'll express it in its simplest radical form as \(\sqrt{19}\).
• In some cases, you might need to split the radical expression using the property \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) to simplify individual parts.

By applying these steps, you've reduced complex radicals to their simplest forms, making them much easier to work with in mathematical operations.

Fractions and Radicals

Fractions involving radicals can seem intimidating at first, but breaking them down into smaller parts makes them more manageable. Consider the fraction \(\sqrt{\frac{a}{b}}\). To simplify this:

• Use the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) to separate the fraction into two individual radicals.
• Simplify each radical separately if possible. For example, in \(\sqrt{\frac{19}{4}}\), \(\sqrt{4}\) becomes 2.
• Combine these results to form a simplified expression, like \(\frac{\sqrt{19}}{2}\).

When working with fractions and radicals, always look for perfect squares to simplify first. This approach reduces complex expressions into simpler and more interpretable forms.