Problem 66 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rationalize the denominator of the expression. $$ \frac{x y}{\sqrt{x}+\sqrt{y}} $$

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{xy(x - \sqrt{y}+\sqrt{x} - \sqrt{xy})}{x-y}\).

Step 1: Multiply by the conjugate

To rationalize the denominator of the expression, multiply both the numerator and denominator by the conjugate of the denominator, \(\sqrt{x} - \sqrt{y}\). $$ \frac{xy}{\sqrt{x}+\sqrt{y}}\times \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} - \sqrt{y}} $$

Step 2: Expanding the numerator

Expand the numerator by distributing the multiplication. $$ \frac{x^2y - xy\sqrt{y} + xy\sqrt{x} - y^2\sqrt{x\sqrt{y}}}{(\sqrt{x}+\sqrt{y})(\sqrt{x} - \sqrt{y})} $$

Step 3: Factor out common terms

Factor out common terms in the numerator. $$ \frac{xy(x - \sqrt{y}+\sqrt{x} - \sqrt{xy})}{(\sqrt{x}+\sqrt{y})(\sqrt{x} - \sqrt{y})} $$

Step 4: Expanding the denominator

Now, we will expand the denominator using the difference of squares theorem. The denominator is the difference of two squares \((a + b)(a - b) = a^2 - b^2\). Here, \(a = \sqrt{x}\) and \(b = \sqrt{y}\). $$ \frac{xy(x - \sqrt{y}+\sqrt{x} - \sqrt{xy})}{(\sqrt{x})^2 - (\sqrt{y})^2} $$

Step 5: Simplifying the expression

Simplify the expression. $$ \frac{xy(x - \sqrt{y}+\sqrt{x} - \sqrt{xy})}{x - y} $$ The given expression, with the denominator rationalized, is: $$ \frac{xy(x - \sqrt{y}+\sqrt{x} - \sqrt{xy})}{x-y} $$

Key Concepts

Understanding the Conjugate

The term 'conjugate' might sound complex, but it's a helpful tool in simplifying mathematical expressions. In essence, the conjugate of a binomial expression like \(a + b\), is \(a - b\). If the expression involves a square root, such as \(\sqrt{x} + \sqrt{y}\), its conjugate would be \(\sqrt{x} - \sqrt{y}\). This process is critical because multiplying an expression by its conjugate removes the square roots in the denominator, thus 'rationalizing' it.
To rationalize the denominator of an expression, we multiply both the numerator and the denominator by the conjugate of the denominator. This process not only simplifies the fraction but makes it more usable in further calculations or simplifications.

Difference of Squares Simplification

The difference of squares is a powerful algebraic tool represented by the formula \(a^2 - b^2 = (a+b)(a-b)\). It simplifies expressions where two perfect squares are subtracted from each other.
In our exercise, the denominator involves terms like \(\sqrt{x} + \sqrt{y}\) and \(\sqrt{x} - \sqrt{y}\). Multiplying these expressions using difference of squares gives us \(x - y\), simplifying our calculation significantly.
This simplification process works because the result of multiplying a sum and a difference of the same terms (conjugates) always results in the difference of the squares of those terms. This method is frequently employed to remove radicals from denominators, as it creates a polynomial without a square root.

Simplifying Expressions

Simplifying mathematical expressions is akin to cleaning up a room - it clears clutter and makes a problem easier to digest. The steps to simplify an expression often involve combining like terms, using properties of arithmetic, and employing algebraic identities such as the difference of squares or the distributive property.
In this exercise, you see simplification in action as we factor out common terms from our expanded numerator. By recognizing common terms and factors, we reduce the complexity of the expression, streamlining it into a more manageable form. This process makes the expression more efficient to work with in computational tasks or further algebraic manipulation.

Navigating Mathematical Expressions

Mathematical expressions are like sentences using a different language. They convey quantities, operations, and relationships. The key to understanding them lies in recognizing familiar structures and patterns.

• Variables and constants represent the core components.
• Operators, such as addition, subtraction, multiplication, and division, build relationships between those components.
• Properties, such as distributive or associative, guide the simplification.
• Special terms and identities, like conjugates or the difference of squares, provide powerful shortcuts for simplifying and solving.

In this rationalization exercise, we navigate these concepts by identifying the appropriate use of conjugates and difference of squares. Once understood, these tools transform often daunting, complex expressions into solvable and interpretable forms. The improved clarity not only aids in solving the current problem but also builds a strong foundation for tackling future mathematical challenges.