Problem 42 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. $$ \frac{2}{\sqrt{2}} $$

Step-by-Step Solution

Answer

The simplification of the given radical expression, \(\frac{2}{\sqrt{2}}\), is \(\sqrt{2}\)

Step 1: Rationalize the denominator

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{2}\). This gives :\[\frac{2}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}}\]

Step 2: Multiply the fractions

After multiplying, you'll have: \[\frac{2*\sqrt{2}}{2}\]

Step 3: Simplify the expression

The number 2 in the numerator and the denominator cancel each other out, leaving: \[\sqrt{2}\]

Key Concepts

Rationalizing the Denominator

When dealing with fractions that have radicals in the denominator, rationalizing helps to eliminate the radical from the bottom of the fraction. This action simplifies the expression and makes it easier for most calculations and interpretations.
Rationalizing involves multiplying both the numerator and the denominator by a conjugate or necessary expression to remove the radical.
In the given example, you start with the fraction \( \frac{2}{\sqrt{2}} \). The denominator has \( \sqrt{2} \), a radical. Multiply both the numerator and the denominator by \( \sqrt{2} \):

• The numerator becomes \( 2 * \sqrt{2} \).
• The denominator becomes \( \sqrt{2} * \sqrt{2} = 2 \).

This leads to a new fraction after the rationalization process. This method helps smoothen calculations, especially when adding or subtracting fractions.

Simplifying Radicals

Simplifying radicals means finding an equivalent expression with the smallest possible radical component. This process can often make mathematical expressions easier to work with and understand.
In the original exercise: \( \frac{2}{\sqrt{2}} \), once rationalized, results in \( \frac{2\sqrt{2}}{2} \).
To simplify a radical expression, follow these steps:

• Observe the fraction \( \frac{2\sqrt{2}}{2} \): here, \( 2 \) in the numerator and the denominator can cancel each other out.
• This cancellation simplifies the entire expression to just \( \sqrt{2} \).

Simplified radicals are easier to work with, reducing calculation complexity in broader mathematical problems.

Mathematical Fractions

Fractions are fundamental components in mathematics, representing a portion of a whole. Simplifying fractions can often make computations more manageable and reveals the core essence of the expression.
In our example, the fraction \( \frac{2}{\sqrt{2}} \) after rationalizing yields \( \frac{2\sqrt{2}}{2} \), which is still a fraction.
Understanding fractions involves:

• Identifying the numerator and the denominator.
• Performing operations like multiplication or division to simplify.
• Noticing when numbers in the numerator and denominator can cancel.

When simplified properly, as in this case, the original fraction \( \frac{2}{\sqrt{2}} \) transforms into the simpler form of \( \sqrt{2} \), no longer a fraction. This process illustrates how fraction manipulation can simplify complex expressions. Understanding these principles makes working with mathematical fractions less daunting and more intuitive.