Problem 27 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The fraction with rationalized denominator is \(\frac{\sqrt{2x}}{2}\)

Step 1: Identify the fraction to be rationalized

The fraction we need to rationalize is \(\frac{\sqrt{x}}{\sqrt{2}}\). We will do this by eliminating the square root from the denominator.

Step 2: Rationalize the denominator

To eliminate the square root from the denominator, we multiply the numerator and the denominator by \(\sqrt{2}\). Putting this into the fraction we get \(\frac{\sqrt{x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}\).

Step 3: Simplify the fraction

We multiply numerator with numerator and denominator with denominator. This gives us \(\frac{\sqrt{2x}}{2}\). Hence the denominator of the fraction has been rationalized.

Key Concepts

Radical Expressions

Radical expressions involve roots, most commonly square roots, depicted by the radical sign (√). When dealing with radical expressions, it's important to understand how radicals interact with each other. The goal, often, is to simplify the expressions or perform operations to make them easier to work with.

For any variable or number under a square root, the radical essentially asks "what number, multiplied by itself, equals what's inside the radical?" For instance, with the expression \( \sqrt{x} \), it's seeking the number that, when squared, equals \( x \).

Rationalizing radicals, particularly in denominators, helps to achieve a form that is simpler or more conventional for further calculations. This is why eliminating square roots from denominators is such a common task in mathematical problems involving radicals.

Simplifying Fractions

When simplifying fractions, the primary aim is to rewrite them in the simplest form. This entails ensuring both the numerator and the denominator have no common factors other than 1. In the context of expressions involving radicals, simplifying fractions often includes rationalizing the denominator.

For instance, in the fraction \( \frac{\sqrt{x}}{\sqrt{2}} \), we simplify it by multiplying both the numerator and denominator by \( \sqrt{2} \). This action clears the square root in the denominator:

• The numerator becomes \( \sqrt{x} \cdot \sqrt{2} = \sqrt{2x} \)
• The denominator transforms to \( \sqrt{2} \cdot \sqrt{2} = 2 \)

Thus, we've achieved the fraction \( \frac{\sqrt{2x}}{2} \), which is simpler and more conventional for further manipulation or evaluation.

Square Roots

Square roots are one of the most basic forms of radicals. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, written as \( \sqrt{9} = 3 \), because \( 3 \times 3 = 9 \).

Understanding square roots is crucial when dealing with radical expressions and rationalizing denominators. In the given exercise, we worked with square roots like \( \sqrt{x} \) and \( \sqrt{2} \).

When rationalizing expressions with square roots, it's often necessary to multiply by a conjugate or the root itself to eliminate the radical in a denominator, thereby simplifying calculations and making further mathematical operations straightforward. This process is key to managing and simplifying expressions in mathematics effectively.