Problem 98 - College Algebra with Modeling and Visualization

Question

Rationalize the denominator. $$ \frac{8}{\sqrt{2}} $$

Step-by-Step Solution

Answer

The rationalized form is \(4\sqrt{2}\).

Step 1: Identify the Conjugate

The denominator of the expression \( \frac{8}{\sqrt{2}} \) is \( \sqrt{2} \). To rationalize the denominator, multiply by the conjugate of the denominator. In this case, since the denominator is \( \sqrt{2} \), the conjugate is simply \( \sqrt{2} \) as well.

Step 2: Multiply the Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator of the fraction by \( \sqrt{2} \): \[ \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}. \]

Step 3: Simplify the Denominator

Simplify the expression \( \sqrt{2} \times \sqrt{2} \). Since \( \sqrt{2}^2 = 2 \), the denominator simplifies to 2.

Step 4: Simplify the Entire Expression

Now calculate the numerator: \( 8 \times \sqrt{2} = 8\sqrt{2} \). Dividing this by the simplified denominator gives: \[ \frac{8\sqrt{2}}{2}. \]

Step 5: Simplify the Result

Simplify \( \frac{8\sqrt{2}}{2} \) by dividing both the coefficient and the denominator by 2: \[ \frac{8\sqrt{2}}{2} = 4\sqrt{2}. \]

Key Concepts

Understanding Square Roots

Square roots are an essential mathematical concept. They represent the number which, when multiplied by itself, results in the original number. For example, the square root of 4 is 2, because \( 2 \times 2 = 4 \). Understanding square roots is crucial in algebra, particularly when dealing with radicals and expressions that involve exponents.

The square root symbol \( \sqrt{} \) is used to denote the square root of a number. In the exercise, the number \( \sqrt{2} \) is part of the denominator in the fraction \( \frac{8}{\sqrt{2}} \).

Here are a few key points to remember about square roots:

• Square roots can be both positive and negative because both give the same square when multiplied by itself.
• When \( n^2 \) equals a number, \( n \) is the square root of that number.
• Square roots play a pivotal role when simplifying algebraic expressions.

Being familiar and comfortable with square roots forms the foundation for more complex algebraic operations.

Working with Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are a way to represent mathematical statements using symbols. Understanding how to manipulate these expressions is vital in solving equations, simplifying expressions, and performing calculations.

In this exercise, \( \frac{8}{\sqrt{2}} \) is an algebraic expression. Here the fraction signifies division and shows the division of 8 by \( \sqrt{2} \). Handling such expressions often involves transforming or simplifying them to a more manageable form.

Some essentials to keep in mind about algebraic expressions:

• Expressions can include operators like addition, subtraction, multiplication, and division.
• Variables are symbols that stand in for numbers in expressions, equations, and inequalities.
• When working with fractions in algebraic expressions, rationalizing the denominator is a common technique, which we see in this exercise.

This familiarity lets you navigate various mathematical problems without getting overwhelmed by complexity.

Simplifying Fractions

Simplifying fractions involves reducing them into their simplest form. This means making the numerator and the denominator as small as possible while keeping their ratio the same. The main goal is to make the expression easier to understand and work with.

In the exercise, the fraction \( \frac{8}{\sqrt{2}} \) went through several steps until it was simplified to \( 4\sqrt{2} \). This process required rationalizing the denominator, which is a technique to eliminate square roots from the denominator.

Here's how simplification works:

• Firstly, rationalize the denominator by multiplying the numerator and the denominator by the square root that's in the denominator.
• This often involves turning expressions like \( \frac{8}{\sqrt{2}} \) to \( \frac{8\sqrt{2}}{2} \).
• Finally, divide both the numerator and denominator by any common factors to bring the fraction to its simplest form, which became \( 4\sqrt{2} \) in this case.

Simplifying fractions not only makes them easier to interpret but also aids in performing further calculations and in achieving a clearer solution.