Problem 68 - Precalculus: Functions and Graphs

Question

Rationalize the numerator. $$\frac{\sqrt{x}-\sqrt{x+h}}{h \sqrt{x} \sqrt{x+h}}$$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{-1}{ \sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} \).

Step 1: Identify the expression

The given expression is \( \frac{\sqrt{x}-\sqrt{x+h}}{h \sqrt{x} \sqrt{x+h}} \). Our goal is to rationalize its numerator, which means we need to eliminate the square roots from it.

Step 2: Multiply by the Conjugate

To rationalize the numerator \( \sqrt{x} - \sqrt{x+h} \), multiply both the numerator and denominator by the conjugate of the numerator, which is \( \sqrt{x} + \sqrt{x+h} \). This gives: \[ \frac{(\sqrt{x} - \sqrt{x+h})(\sqrt{x} + \sqrt{x+h})}{h \sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} \]

Step 3: Apply the Difference of Squares Formula

Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we simplify the numerator: \( (\sqrt{x})^2 - (\sqrt{x+h})^2 = x - (x+h) = -h \). So now, the expression becomes: \[ \frac{-h}{h \sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} \]

Step 4: Simplify the Expression

Notice that \( h \) in the numerator and the denominator cancels out: \[ \frac{-h}{h \sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} = \frac{-1}{ \sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} \] Now, the numerator is rationalized.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that helps simplify expressions involving the subtraction of two square roots. When applied to expressions like \( \sqrt{x} - \sqrt{x+h} \), it allows us to eliminate square roots by transforming the expression into a more manageable form.

Using the difference of squares formula, \( (a - b)(a + b) = a^2 - b^2 \), we simplify the multiplicative product of the binomial and its conjugate. Here, \( a \) and \( b \) are \( \sqrt{x} \) and \( \sqrt{x+h} \), respectively. When we multiply \( (\sqrt{x} - \sqrt{x+h})(\sqrt{x} + \sqrt{x+h}) \), we obtain \( (\sqrt{x})^2 - (\sqrt{x+h})^2 \).

This results in us having \( x - (x + h) = -h \). Applying this identity simplifies complex expressions by removing the radicals, making it easier to proceed with further simplification.

Conjugate Multiplication

Conjugate multiplication is a tactic used to rationalize expressions, particularly useful when dealing with square roots in the numerator or denominator of fractions. The trick is to multiply the expression by a version of itself, known as its conjugate, which alters how the equation is structured.

For example, the conjugate of an expression \( \sqrt{x} - \sqrt{x+h} \) is \( \sqrt{x} + \sqrt{x+h} \). By multiplying the original expression and its conjugate, the radicals in the numerator are effectively neutralized through the difference of squares method.

• Original expression: \( \sqrt{x} - \sqrt{x+h} \)
• Conjugate: \( \sqrt{x} + \sqrt{x+h} \)

Once the conjugate is applied, the numerator loses the square roots, simplifying the problem substantially. This technique is especially handy when rationalizing numerators, as it paves the way for further cancellation of terms and simplification.

Canceling Terms

Canceling terms is a vital step in simplifying rational expressions. After applying the difference of squares and conjugate multiplication, we often find common factors in the numerator and the denominator, which can be canceled out to simplify the expression further.

In our example, after rationalizing the numerator, we have \( \frac{-h}{h \sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} \). Here, \( h \) appears in both the numerator and the denominator. By canceling it out, we simplify the expression to \( \frac{-1}{\sqrt{x} \sqrt{x+h} (\sqrt{x} + \sqrt{x+h})} \).

• Identify common factors in the numerator and denominator.
• Cancel these terms to reduce the expression.

This step is crucial as it not only simplifies the expression but also prepares it for further evaluation or final result interpretation.

Rational Expressions

Rational expressions involve fractions where the numerator and/or the denominator are polynomials. Working with these expressions requires specific techniques, particularly when they include complex operations like square roots.

Rationalizing a fraction, as seen in the original exercise, simplifies the expression by eliminating radicals from the numerator. This process helps in evaluating or simplifying complex rational expressions efficiently.

• Identify the part of the expression that needs rationalizing.
• Apply suitable algebraic identities like the difference of squares.
• Simplify by canceling common terms.

Understanding and manipulating rational expressions is a critical skill in algebra that enhances problem-solving capabilities by streamlining complex calculations.