Problem 5 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{4}{\sqrt{2}}\)

Step-by-Step Solution

Answer

The simplified form is \( 2\sqrt{2} \).

Step 1: Identify the need to rationalize

Rationalizing the denominator involves removing the square root from the denominator of the fraction. In this case, the given fraction is \( \frac{4}{\sqrt{2}} \). The denominator is \( \sqrt{2} \), which is an irrational number.

Step 2: Multiply numerator and denominator by the square root

To remove the square root from the denominator, multiply both the numerator and the denominator by \( \sqrt{2} \). This gives:\[ \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} \].

Step 3: Simplify the expression

Now simplify the fraction. The denominator becomes \( 2 \) because \( \sqrt{2} \times \sqrt{2} = 2 \). The fraction simplifies to \( \frac{4\sqrt{2}}{2} = 2\sqrt{2} \).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying is essential to make them as neat and manageable as possible. A fraction is considered simplified when the numerator and the denominator do not have any common factors other than 1. Simplifying fractions makes calculations easier and helps you understand the fraction's value more clearly.

To simplify a fraction, you

• find the greatest common divisor (GCD) of the numerator and the denominator
• then divide both the numerator and the denominator by this number

For example, consider a simple fraction, like \( \frac{6}{8} \). The greatest common divisor for 6 and 8 is 2. Dividing both the numerator and the denominator by 2, you get \( \frac{3}{4} \). It is now in its simplest form.

Although rationalizing the denominator, as shown in the original exercise, is a specialized case of simplifying, it clarifies the expression further by ensuring the denominator is free of irrational numbers. This brings us to the next concept.

Irrational Numbers

Understanding irrational numbers is crucial in mathematics, especially in topics like rationalizing. An irrational number cannot be expressed as a simple fraction, as it doesn't end or repeat. Think of numbers like \( \pi \) or \( \sqrt{2} \) - these are irrational.

Irrational numbers are significant in various mathematical contexts. Since they cannot be precisely expressed as a ratio of two integers, they often create complexity in equations and expressions.

When dealing with fractions that have irrational numbers in the denominator, like \( \frac{4}{\sqrt{2}} \), rationalizing becomes necessary. This process involves altering the expression to ensure no irrational number stays in the denominator. It helps make calculations easier and expressions more meaningful.

Square Roots

Square roots might seem daunting, but once you understand them, they become straightforward. A square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For instance, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).

In many instances, especially in algebra, you'll encounter square roots of non-perfect squares. These usually result in irrational numbers. Consider \( \sqrt{2} \). It's an irrational number because it cannot be written as a simple fraction, and its decimal representation never ends nor repeats.

When you have a square root in the denominator, like in the exercise with \( \sqrt{2} \), it complicates computations. This is where rationalizing the denominator becomes necessary. By multiplying the fraction by a form of 1 that includes the square root, such as \( \frac{\sqrt{2}}{\sqrt{2}} \), you can eliminate the irrational number from the denominator and simplify your expression.