Problem 14 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \frac{\sqrt{7}}{\sqrt{28}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \).

Step 1: Rationalizing the denominator

To simplify the expression \( \frac{\sqrt{7}}{\sqrt{28}} \), we can start by rationalizing the denominator. Notice that \( \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \). Therefore, the expression becomes \( \frac{\sqrt{7}}{2\sqrt{7}} \).

Step 2: Simplifying the fraction

Now that we have \( \frac{\sqrt{7}}{2\sqrt{7}} \), we can simplify this fraction by canceling out the \( \sqrt{7} \) terms in the numerator and the denominator. This leaves us with \( \frac{1}{2} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is an essential skill when dealing with fractions that contain radical expressions in the denominator. This process makes calculations and comparisons easier, avoiding the complication of radicals in the denominator.
To rationalize, you essentially "get rid" of the square root by multiplying both the numerator and the denominator by a number that will make the denominator a perfect square.

In our exercise, we have the fraction \( \frac{\sqrt{7}}{\sqrt{28}} \).
- Start by expressing 28 as a product of 4 and 7: \( \sqrt{28} = \sqrt{4 \times 7} \).
- Simplifying this gives \( 2\sqrt{7} \). Thus, we rewrite the expression as \( \frac{\sqrt{7}}{2\sqrt{7}} \).

This step helps keep mathematical expressions in a more conventional and widely accepted format, which is especially useful in advanced math topics.

Fraction Simplification

Simplifying fractions is a fundamental part of working with all types of expressions and is not confined just to numeric fractions. Fraction simplification involves making the fraction as simple as possible without changing its value.

For the expression \( \frac{\sqrt{7}}{2\sqrt{7}} \), simplification involves:

• Identifying common factors in the numerator and the denominator.
• Canceling out these common terms.

Here, both the numerator (\( \sqrt{7} \)) and the denominator (\( 2\sqrt{7} \)) include \( \sqrt{7} \).
- By dividing both by \( \sqrt{7} \), we obtain \( \frac{1}{2} \).

This action reduces the expression to its simplest form, making it easier to interpret and apply.

Radical Expressions

Radical expressions include any expression that contains a square root, cube root, or other mathematical root. Simplifying radical expressions is important for making them easier to work with,
and it often involves rationalizing the denominator or removing factors from under the root.

In our exercise, the radical expression is \( \frac{\sqrt{7}}{\sqrt{28}} \).
- Recognize that \( \sqrt{28} \) can be simplified.
- Simplifying \( \sqrt{28} \) to \( 2\sqrt{7} \) transforms the original expression into a simpler form.

This simplification helps us work with the fraction more effectively, leading to the simplified result of \( \frac{1}{2} \). Understanding the structure and properties of radical expressions is key to manipulating and simplifying them correctly.