Problem 35 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\frac{2y\sqrt{x} + 2x\sqrt{y}}{xy}\)

Step 1: Rationalize the First Fraction

To rationalize the denominator of the first fraction \(\frac{2}{\sqrt{x}}\), multiply both the numerator and the denominator by \(\sqrt{x}\). This gives:\[\frac{2}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{2\sqrt{x}}{x}.\]

Step 2: Rationalize the Second Fraction

To rationalize the denominator of the second fraction \(\frac{2}{\sqrt{y}}\), multiply both the numerator and the denominator by \(\sqrt{y}\). This results in:\[\frac{2}{\sqrt{y}} \times \frac{\sqrt{y}}{\sqrt{y}} = \frac{2\sqrt{y}}{y}.\]

Step 3: Combine the Rationalized Fractions

Now that both fractions have been rationalized, combine them over a common denominator. The least common denominator (LCD) is \(xy\). Rewriting the fractions over \(xy\) gives:\[\frac{2\sqrt{x}}{x} + \frac{2\sqrt{y}}{y} = \frac{2\sqrt{x} \cdot y}{xy} + \frac{2\sqrt{y} \cdot x}{xy}.\]

Step 4: Simplify the Combined Fraction

Combine the numerators over the common denominator:\[\frac{2y\sqrt{x} + 2x\sqrt{y}}{xy}.\]This is the simplest form with a rationalized denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions is one of the foundational concepts in mathematics. It involves reducing fractions to their simplest or lowest terms. This makes calculations easier and results more understandable.

To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. This process helps eliminate unnecessary complexity in fractions by reducing their size without changing their value.

Here are some tips on simplifying fractions:

• Identify the GCD of the numerator and denominator.
• Divide both by this number.
• Continue to simplify until the GCD is 1.

Simplifying fractions is crucial before performing any arithmetic operations, as it ensures precise results and eases further calculations.

Common Denominator

When working with fractions, especially in addition and subtraction, it's important to have a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. Having a common denominator allows you to easily combine the fractions.

In our exercise, after rationalizing the denominators of each fraction, the next step is combining them which requires finding a common denominator. Here, the least common denominator (LCD) is used. The LCD is the smallest number that each of the denominators can divide evenly. This ensures both fractions can be rewritten to have the same denominator, thereby facilitating their addition or subtraction.

Steps to find a common denominator:

• List the multiples of each denominator.
• Identify the least common multiple (LCM) that appears in both lists.
• Rewrite each fraction with this common denominator.

Finding a common denominator is vital for mixing fractions efficiently, allowing for seamless arithmetic operations.

Square Roots

Square roots are related to the concept of finding a number which, when multiplied by itself, gives the original number. The square root symbol, \( \/ \ \ \ \), indicates this operation.

In the context of rationalizing denominators, square roots often appear in the denominator, like in the fractions \( \/ \ \ x \ \ \ \) and \( \/ \ \ y \ \ \ \). Rationalizing involves eliminating the square root from the denominator by multiplying the fraction by a form of 1 that contains the square root in the numerator and denominator. This results in a fraction with the denominator being a whole number.

Understanding square roots in rationalizing fractions allows us to analyze and simplify fractions effectively, helping students solve more complex problems involving irrational numbers and algebraic expressions.