Problem 93 - College Algebra with Modeling and Visualization

Question

Multiply and simplify. $$ (\sqrt{a b}-\sqrt{c})(\sqrt{a b}+\sqrt{c}) $$

Step-by-Step Solution

Answer

The simplified expression is \(ab - c\).

Step 1: Recognize the Product of Conjugates

Identify the expression \[(\sqrt{ab} - \sqrt{c})(\sqrt{ab} + \sqrt{c})\]as a product of conjugates. For expressions of the form \((x - y)(x + y)\),the result is a difference of squares, i.e., \(x^2 - y^2\).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula to the expression:\[(\sqrt{ab})^2 - (\sqrt{c})^2\]

Step 3: Simplify Each Term

Calculate each term separately. First, \((\sqrt{ab})^2 = ab\),and then,\((\sqrt{c})^2 = c\).

Step 4: Subtract

Combine the terms from the previous step to complete the simplification:\[ab - c\].

Key Concepts

Conjugate pairs

Conjugate pairs are expressions that involve two binomials, with the same terms but opposite signs. For example, in \((x - y)(x + y)\), the terms are \(x\) and \(y\), and the expressions differ by their signs, namely a minus and a plus.

Conjugate pairs are useful because they simplify the multiplication process. When multiplied,

• They follow the format of \((a - b)(a + b)\).
• The product is a difference of squares, written as \(a^2 - b^2\).

The unique characteristic of conjugate pairs is that the middle terms cancel each other out. This leaves only two terms to subtract from each other, simplifying the expression considerably. In algebra, recognizing conjugate pairs is a key strategy in simplifying complex expressions.

Difference of squares

A difference of squares is an algebraic technique used to simplify expressions that are the result of multiplying conjugate pairs. It is important because it reduces binomial expressions to their simplest form with minimal effort.

The formula to use is straightforward:

• \((x - y)(x + y) = x^2 - y^2\).

This formula can transform problems that seem complicated into much simpler single-step subtractions. Applying this to an expression involves raising each term of the binomials to the power of two and then subtracting the results.

In the given exercise, we had \((\sqrt{ab} - \sqrt{c})(\sqrt{ab} + \sqrt{c})\). Recognizing it as a difference of squares, it becomes \( (\sqrt{ab})^2 - (\sqrt{c})^2 \), which simplifies to \(ab - c\). This technique not only simplifies multiplication but also helps unfold the hidden structure within a problem.

Simplification of radicals

Simplifying radicals involves expressing a square root or other root expression in its simplest form. In the simplification process, we aim to break down a radical into its most basic elements, to make the expression easier to work with in algebraic equations.

The key steps

• Simplifying each component of the radical expression.
• Simplifying using mathematical rules, such as \((\sqrt{x})^2 = x\).

In our example, \((\sqrt{ab})^2\) simplifies to \(ab\), and \((\sqrt{c})^2\) simplifies to \(c\). Then, by subtracting these simplified values, we obtained the expression \(ab - c\).

Identifying and applying these simplification techniques of radicals makes larger expressions much more manageable. Simplification is critical in solving and understanding more complex equations.