Problem 66 - Precalculus: Functions and Graphs

Question

Rationalize the numerator. $$\frac{\sqrt{b}+\sqrt{c}}{b^{2}-c^{2}}$$

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{1}{(b+c)(\sqrt{b} - \sqrt{c})}\).

Step 1: Identifying the Structure

To rationalize the numerator, we need to eliminate the square root in the numerator \(\sqrt{b} + \sqrt{c} \). The key is to use the conjugate of the numerator.

Step 2: Multiply by the Conjugate

The conjugate of the numerator \(\sqrt{b} + \sqrt{c}\) is \(\sqrt{b} - \sqrt{c}\). Multiply both the numerator and denominator of the expression by this conjugate:\[\frac{(\sqrt{b} + \sqrt{c})(\sqrt{b} - \sqrt{c})}{(b^2-c^2)(\sqrt{b} - \sqrt{c})}\]

Step 3: Simplifying the Numerator

Use the formula \((a+b)(a-b) = a^2-b^2\) to simplify the numerator:\[(\sqrt{b})^2 - (\sqrt{c})^2 = b - c\]

Step 4: Complete the New Expression

Substitute the simplified numerator back into the expression:\[\frac{b-c}{(b^2-c^2)(\sqrt{b} - \sqrt{c})}\]

Step 5: Simplifying the Denominator (if necessary)

Simplify the denominator using the difference of squares formula if possible: \[b^2 - c^2 = (b+c)(b-c)\]Substitute this into the denominator:\[\frac{b-c}{(b+c)(b-c)(\sqrt{b} - \sqrt{c})}\]

Step 6: Cancel Common Factors

Since \(b-c\) is a factor in both the numerator and denominator, cancel out \(b-c\), assuming \(b eq c\), to simplify the expression:\[\frac{1}{(b+c)(\sqrt{b} - \sqrt{c})}\]

Key Concepts

Conjugate

Rationalizing the numerator often involves using something called the conjugate. In mathematics, the conjugate of an expression like \( \sqrt{b} + \sqrt{c} \) is its counterpart obtained by changing the sign in the middle: \( \sqrt{b} - \sqrt{c} \). This technique is helpful because, when multiplied, these conjugate pairs eliminate the square roots, which is usually the main goal.

• Conjugates reflect symmetry in mathematical expressions, as one mirrors the other.
• They are especially useful in rationalizing numerators or denominators.

Consider an example: multiplying \( (\sqrt{b} + \sqrt{c}) \) by \( (\sqrt{b} - \sqrt{c}) \) leads to \( b - c \). This result comes from the elimination of the square roots, thanks to their conjugate property. This makes further simplification of complex expressions possible.

Difference of Squares

The "difference of squares" is a formula frequently used in mathematics:\((a^2 - b^2 = (a+b)(a-b))\). Understanding this formula will significantly aid in recognizing and simplifying expressions. For example, when you see an expression like \( b^2 - c^2 \), immediately think of the difference of squares. Substitute with its factored form: \((b + c)(b - c)\).

• This factorization can simplify multiplication and division procedures in algebra.
• Recognizing it enables faster manipulation and simplification of algebraic expressions.

This property was used in simplifying the denominator \( b^2 - c^2 \) in the original problem, resulting in well-formed factored expressions that pave the way for cancellations.

Simplification

Simplification in mathematics is the process of reducing complexity, often making expressions easier to work with or understand. In algebra, it involves several methods such as factoring, cancelling common factors, or applying special formulas.When you multiply the numerator and denominator by the conjugate and use the difference of squares, the expression simplifies to remove roots.

• This process often involves substitution, like in the original solution substituting \(b^2 - c^2\) with \((b+c)(b-c)\).
• By cancelling out common factors, like \(b-c\), the expression becomes simpler and easier to interpret.

In general, simplification aims at reducing expressions into their simplest forms, which reveals their true structure without losing value.

Algebraic Manipulation

Algebraic manipulation involves the use of algebraic techniques to transform mathematical expressions into desired forms. This can involve operations such as addition, subtraction, multiplication, division, and factoring among others. In the original problem, we used algebraic manipulation when multiplying by the conjugate and simplifying complex expressions.

• The aim is often to solve equations or simplify for readability and ease of use.
• This process lets you explore alternative representations, recognize patterns, and find simpler solutions.

Developing competence in algebraic manipulation is crucial for tackling more advanced mathematical problems, where such techniques are fundamental tools for problem-solving.