Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{1}{\sqrt{2}}$$

Short answer

\( \frac{\sqrt{2}}{2} \)

Step-by-step solution

Identify the denominator

The denominator of the expression is \( \sqrt{2} \).

Multiply numerator and denominator by the conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{2} \). The expression becomes: \( \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \).

Simplify the expression

Simplify the resulting expression. The numerator is \( \sqrt{2} \) and the denominator is \( \sqrt{2} \cdot \sqrt{2} = 2 \). Hence, the expression becomes \( \frac{\sqrt{2}}{2} \).

Key concepts

simplifying expressions

One of the essential skills in mathematics is simplifying expressions. Simplifying means reducing an expression to its simplest form, where it is easier to work with. This usually involves:

• Combining like terms
• Reducing fractions
• Using mathematical properties and operations, such as the distributive property, to make an expression simpler

For example, in our problem, we start with \(\frac{1}{\frac{\text{√}{2}}}\). Multiplying both the numerator and denominator by \(\text{√}{2}\) helps to rationalize the denominator. When we rationalize the denominator, we are making the expression easier to handle by eliminating the square root in the denominator. Remember, rationalizing an expression keeps its value the same; we are just rewriting it in a simpler form.

square roots

Square roots are an essential concept in algebra and often appear in various mathematical contexts. The square root of a number \(n\) is a value that, when multiplied by itself, gives \(n\). In our exercise, we have \(\text{√}{2}\) in the denominator. The goal is to eliminate the square root in the denominator. Multiplying the fraction by \( \frac{\text{√}{2}}{\text{√}{2}} \) effectively changes the denominator from \(\text{√}{2}\) to 2 because \( \text{√}{2} \) times \( \text{√}{2} \) equals \( 2 \). This concept is critical in rationalizing denominators.

fractions

Fractions represent part of a whole and are written as \( \frac{a}{b} \), where \(a\) is the numerator and \(b\) is the denominator. In our problem, the fraction \( \frac{1}{\text{√}{2}} \) had a square root in the denominator. By multiplying both parts of the fraction by \( \text{√}{2} \), we get \( \frac{\text{√}{2}}{2} \). This new fraction is simplified. Understanding fractions and their properties is crucial when working with expressions involving rationalizing denominators. Key things to remember about fractions:

• Multiplying or dividing both numerator and denominator by the same number doesn't change the fraction's value.
• The goal of simplifying fractions is to rewrite them in a form where they are easier to understand and use in further computations.