Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{5}{\sqrt{29}}$$

Step-by-Step Solution

Answer

\( \frac{5\sqrt{29}}{29} \)

Step 1: Identify the need for rationalization

The given expression \( \frac{5}{\sqrt{29}} \) has an irrational denominator, which is \( \sqrt{29} \). To rationalize this denominator, we need to eliminate the square root from it.

Step 2: Multiply numerator and denominator by the conjugate

To eliminate the square root from the denominator, we multiply both the numerator and the denominator of the fraction by \( \sqrt{29} \). This step does not change the value of the fraction since we are essentially multiplying by 1. This results in: \( \frac{5}{\sqrt{29}} \times \frac{\sqrt{29}}{\sqrt{29}} = \frac{5 \cdot \sqrt{29}}{\sqrt{29} \cdot \sqrt{29}} \).

Step 3: Simplify the expression

Simplify the denominator: \( \sqrt{29} \times \sqrt{29} = 29 \). Thus, the expression now is \( \frac{5\sqrt{29}}{29} \). This is the rationalized form of the original expression.

Key Concepts

Understanding Square Roots

Square roots are essential in mathematics and are denoted by the radical symbol (\( \sqrt{} \)). When you square a number, you are multiplying it by itself, and the square root is the opposite process: finding a number that, when multiplied by itself, gives you the original number.
For example, \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \).
When dealing with expressions, square roots often appear in denominators, creating irrational denominators. An irrational number cannot be expressed as a simple fraction, so we aim to transform it into a fraction, which involves a process called rationalizing the denominator.

• The purpose of rationalizing is to make it easier to perform arithmetic operations, particularly addition and subtraction, with fractions that include square roots.
• For instance, in the example \( \frac{5}{\sqrt{29}} \), \( \sqrt{29} \) is irrational.
• We can rationalize this by multiplying the numerator and denominator by \( \sqrt{29} \), as shown in the steps of the problem.

Exploring Fractions in Detail

Fractions are a way to express numbers that are not whole numbers. They are written with a numerator (the top part) and a denominator (the bottom part). For instance, in the fraction \( \frac{5}{29} \), 5 is the numerator, and 29 is the denominator.
This structure is very useful for expressing numbers between integers and for working with parts of a whole. When dealing with fractions that include expressions such as square roots, it's important to ensure that the form of the fraction is simple and easy to handle.

• If a fraction has an irrational denominator, as in \( \frac{5}{\sqrt{29}} \), we rationalize it by eliminating the square root from the denominator.
• This process involves multiplying both the numerator and the denominator by the same square root, which effectively turns the denominator into a rational number since \( \sqrt{29} \times \sqrt{29} = 29 \).
• This allows the fraction to be expressed in a simplified form, making further operations more straightforward.

Simplifying Expressions Involving Roots and Fractions

Simplifying expressions is a fundamental skill in algebra, especially when they involve roots and fractions. Simplification makes expressions more manageable and easier to work with.
For the expression \( \frac{5}{\sqrt{29}} \), by multiplying the numerator and the denominator by \( \sqrt{29} \), you effectively simplify the expression.

• This process involves removing the square root from the denominator, a crucial step when working with fractions in algebra.
• The denominator becomes \( 29 \) since \( \sqrt{29} \times \sqrt{29} = 29 \), resulting in \( \frac{5 \sqrt{29}}{29} \).
• Simplified expressions like this are inherent to solving algebra problems, allowing for clearer calculations and solutions.

Each step in simplifying expressions involves understanding how to handle radicals, rational numbers, and ensuring that your work is both accurate and readable.