Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\)

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Identify the Expression Type

The given expression \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\) is a product of two binomials in the form of \((a+b)(a-b)\). This is a difference of squares problem.

Step 2: Apply Difference of Squares Formula

Recall that \((a+b)(a-b) = a^2 - b^2\). In our expression, \(a = \sqrt{5}\) and \(b = \sqrt{3}\). Applying this formula gives: \((\sqrt{5})^2 - (\sqrt{3})^2\).

Step 3: Simplify Each Term

Calculate \((\sqrt{5})^2 = 5\) and \((\sqrt{3})^2 = 3\). The expression now becomes \(5 - 3\).

Step 4: Perform the Final Calculation

Subtract to simplify the expression: \(5 - 3 = 2\).

Key Concepts

Difference of Squares

The difference of squares is a unique pattern found in algebra that simplifies expressions of the form \((a+b)(a-b)\). This pattern is incredibly useful because it allows us to transform and simplify expressions quickly without multiplying every term separately. The difference of squares relies on the identity:

• \((a+b)(a-b) = a^2 - b^2\)

This means that when you multiply a sum and a difference of the same two terms, you end up subtracting squares. This is a very efficient way to simplify certain expressions.
In the exercise we looked at, we recognized that the expression \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\) matched the difference of squares pattern. By identifying that \(a = \sqrt{5}\) and \(b = \sqrt{3}\), we were able to directly apply the formula without further lengthy calculations. This required only squaring \(a\) and \(b\), then subtracting. This shortens the process significantly when working through algebra problems.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and beyond. Handling radical expressions becomes straightforward when you understand the rules of exponents and roots. The basics involve:

• Recognizing and simplifying square roots into smaller perfect squares
• Ensuring to rationalize denominators, when necessary

In this exercise, we worked with square roots: \(\sqrt{5}\) and \(\sqrt{3}\).
The square root function essentially asks which number, when multiplied by itself, results in the given number. For example, \((\sqrt{5})^2 = 5\) because multiplying 5 by itself gives 5, similarly for \((\sqrt{3})^2 = 3\). Handling these correctly helps to simplify radical expressions, making problems more manageable or sometimes eliminating radicals altogether.
Perfect understanding of how to manipulate these can easily help you simplify expressions in algebra, a crucial skill.

Binomial Multiplication

Binomial multiplication involves multiplying expressions that contain two terms. These can present themselves in pairs, like \((a+b)(a-b)\). These are essential building blocks for algebraic expressions, allowing us to expand and simplify complex expressions. Each term in one binomial must multiply each term in the other binomial, which can be done easily by:

• Using distributive properties (FOIL method for binomials)
• Applying known identities, like the difference of squares

For instance, multiplying \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\) can initially seem daunting. However, recognizing it as a difference of squares makes the task straightforward, reducing it to squaring terms individually and subtracting them.
By relying on identities and simplification techniques in algebra, such problems become less complex and easier to solve, enhancing problem-solving skills and boosting confidence in handling algebraic equations.