Problem 79 - A Graphical Approach to College Algebra

Question

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{8}{\sqrt{5}}$$

Step-by-Step Solution

Answer

The expression \( \frac{8}{\sqrt{5}} \) is rationalized to \( \frac{8\sqrt{5}}{5} \).

Step 1: Understanding the Problem

We need to rationalize the denominator of the expression \( \frac{8}{\sqrt{5}} \). This means we should eliminate the square root from the denominator by converting it into a rational number.

Step 2: Multiply by a Conjugate

Since the denominator is a single square root \( \sqrt{5} \), we multiply both the numerator and the denominator by \( \sqrt{5} \) to remove the square root. This is done to create a perfect square under the square root in the denominator.\[ \frac{8}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \]

Step 3: Simplify the Expression

After multiplying, the expression becomes:\[ \frac{8 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{8\sqrt{5}}{5} \]Now the denominator becomes rational (5) as \( \sqrt{5} \cdot \sqrt{5} = 5 \).

Step 4: Final Result

The rationalized expression of \( \frac{8}{\sqrt{5}} \) is \( \frac{8\sqrt{5}}{5} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, like square roots, cube roots, or higher. In simple terms, a radical is a mathematical symbol that asks for the root of a number. The most commonly encountered root is the square root, symbolized as \( \sqrt{} \). When you see a radical expression such as \( \sqrt{9} \), it asks for the square root of 9, which is 3. These expressions are important in algebra as they represent precise values that can be simplified but often aren't whole numbers.

Rationalizing radicals is a technique used to eliminate radicals from the denominator. This is important because having radicals in the denominator can complicate calculations, so rationalizing the expression simplifies it and makes it easier to work with. This involves multiplying by a conjugate or another expression that will cancel out the radical when multiplied.

Square Roots

Square roots specifically deal with finding a number which, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \( 5 \times 5 = 25 \).

Understanding square roots is crucial when dealing with radical expressions, especially when rationalizing denominators. In our exercise, the expression \( \frac{8}{\sqrt{5}} \) contains a square root in the denominator, which we aim to eliminate. By multiplying the numerator and the denominator by \( \sqrt{5} \), we turn \( \sqrt{5} \times \sqrt{5} \) into 5, a rational number.

This action simplifies the denomination while keeping the equality of the fraction, illustrating how square roots interplay in rationalizing denominators.

Mathematical Proofs

Mathematical proofs involve demonstrating with logical reasoning why a particular mathematical statement is true. Proofs often require using known formulas and properties. For rationalizing denominators, a proof shows that multiplying by the square root's value over itself ensures that the original expression simplifies correctly.

When we multiply \( \frac{8}{\sqrt{5}} \) by \( \frac{\sqrt{5}}{\sqrt{5}} \), we apply the property that any number divided by itself is 1. This doesn't change the value of the expression but enables us to transform \( \sqrt{5} \) into 5 through multiplication. This algebraic step proves why and how rationalizing changes the form but not the value of an expression.

Algebraic Techniques

Algebraic techniques are methods used in algebra to manipulate mathematical expressions and equations. Rationalizing denominators is just one of many such techniques. It’s used to remove radicals from denominators by strategically multiplying the numerator and denominator by an expression that will simplify the denominator.

The core technique here is to multiply the expression by \( \frac{\sqrt{5}}{\sqrt{5}} \), which is essentially multiplying by 1. This technique is effective because it uses the properties of radicals to transform irrational parts into rational numbers, simplifying calculations.

This makes algebra more manageable, allowing clearer, simpler expressions that are easier to work with in further calculations or problem-solving. Understanding and applying these techniques are fundamental parts of mastering algebra and engaging with complex mathematical problems.