Problem 50 - Contemporary Precalculus

Question

Rationalize the denominator and simplify your answer. $$\frac{\sqrt{x}}{\sqrt{x}-\sqrt{c}}$$

Step-by-Step Solution

Answer

Answer: The rationalized form of the expression is \(\frac{x}{x-c}\).

Step 1: Multiply by the conjugate

We will multiply the numerator and denominator by the conjugate of the denominator which is \(\sqrt{x} + \sqrt{c}\). $$ \frac{\sqrt{x}}{\sqrt{x}-\sqrt{c}} \cdot \frac{\sqrt{x}+\sqrt{c}}{\sqrt{x}+\sqrt{c}} $$

Step 2: Apply the difference of squares

Now, we will compute the numerator and denominator by applying the difference of squares, which is \((a-b)(a+b)=a^2-b^2\). The numerator will be the product of the square roots, and the denominator will be the difference of their squares. $$ \frac{(\sqrt{x})^2}{(\sqrt{x})^2-(\sqrt{c})^2} $$

Step 3: Simplify the expression

Now we will simplify the numerator and denominator. $$ \frac{x}{x-c} $$ So, the final expression is \(\frac{x}{x-c}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics that makes equations more understandable and easier to work with. When we simplify, our goal is to reduce the expression to its simplest form without changing its value. This is particularly important when working with fractions or algebraic expressions.
Simplified expressions can reveal more about the properties of the expression itself and often make subsequent calculations more straightforward.

• Begin by identifying parts of the expression that can be combined or canceled out.
• Look for common factors in the numerator and the denominator if working with fractions.
• Apply mathematical rules to reduce complexity, such as simplifying radicals or combining like terms.

In the exercise, the expression \(\frac{x}{x-c}\) is a simplified form that resulted from eliminating radicals through rationalization. Simplification helps focus on the main elements of the equation and aids further problem solving.

Conjugate of a Denominator

The conjugate of a denominator is an ingenious tool used to rationalize denominators involving square roots. By multiplying the original expression by the conjugate, which is the expression with the opposite sign between terms, we can eliminate radicals from the denominator.
In the given problem, the denominator is \(\sqrt{x} - \sqrt{c}\). Its conjugate is \(\sqrt{x} + \sqrt{c}\). By multiplying the numerator and denominator by this conjugate, the denominator becomes a nice expression that does not include a square root.

• The conjugate of \(a - b\) is \(a + b\), and vice versa.
• Always use the conjugate when you want to eliminate square roots or irrational numbers from the denominator.

This process not only rationalizes the expression but can also lead to more opportunities for simplification in subsequent steps.

Difference of Squares

The difference of squares is a mathematical identity that states \((a-b)(a+b) = a^2 - b^2\). This identity is incredibly useful when dealing with expressions that resemble this format. It allows us to eliminate more complex elements like radicals or certain binomials by transforming them into differences of squares, which are much simpler to handle.
In the exercise, the denominator \((\sqrt{x} - \sqrt{c})(\sqrt{x} + \sqrt{c})\) becomes \(x - c\) after applying this identity. This transformation removes the square roots and leaves a polynomial difference, which is simpler and more desirable.

• Use the difference of squares to break down quadratic expressions or when aiming to simplify products involving conjugates.
• This identity only works when we have a product of two conjugates, making it a perfect match for rationalizing denominators.

Understanding and applying the difference of squares is a powerful technique for efficiently simplifying mathematical expressions that initially appear complex.