Problem 72 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})$$

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Recognize the Pattern

The given expression \((\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})\) is in the form \((a+b)(a-b)\), known as the difference of squares formula. This formula states that \((a+b)(a-b) = a^2 - b^2\).

Step 2: Identify the Components

Identify the components of the expression to apply the formula: \(a = \sqrt{5}\) and \(b = \sqrt{2}\).

Step 3: Apply the Difference of Squares Formula

Using the differences of squares formula, \((a+b)(a-b) = a^2 - b^2\), substitute \(a = \sqrt{5}\) and \(b = \sqrt{2}\). This leads to the expression \(\sqrt{5}^2 - \sqrt{2}^2\).

Step 4: Simplify the Expression

Calculate \(\sqrt{5}^2\) and \(\sqrt{2}^2\). Since squaring reverses the operation of taking a square root, we have \(\sqrt{5}^2 = 5\) and \(\sqrt{2}^2 = 2\).

Step 5: Final Calculation

Subtract the two results obtained: \(5 - 2 = 3\).

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to reduce them to their simplest form, making them easier to work with. The expression given in this exercise, \((\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})\), utilizes the difference of squares, which is a key strategy in simplifying. To apply this concept, we need to identify the components that fit the pattern \((a+b)(a-b)\), which simplifies to \(a^2 - b^2\).

• Start by recognizing the specific pattern that allows simplification.
• Identify each term in the pattern, here being \(a = \sqrt{5}\) and \(b = \sqrt{2}\).
• Substitute directly into the simplified formula, yielding \(\sqrt{5}^2 - \sqrt{2}^2\).

Remember that the main purpose of this simplification is to condense the expression into something more straightforward, in this case from a product of two expressions to a simple subtraction, yielding \(3\). This makes further mathematical operations easier and more efficient.

Real Numbers

Real numbers include all the numbers on the number line, encompassing both the rational and irrational numbers. In this problem, we specifically work with nonnegative real numbers. Here are some important points regarding real numbers:

• Real numbers can be positive, negative, or zero, though in our exercise, we focus on nonnegative values.
• Rational numbers are real numbers that can be expressed as fractions. Irrational numbers, like \(\sqrt{2}\) and \(\sqrt{5}\), cannot be expressed precisely as a fraction.
• All numeric values that do not fall under imaginary numbers are considered real. They help quantify continuous quantities on the number line.

In simplifying the expression, we only deal with nonnegative real numbers, emphasizing the efficiency of the implementation of the formula without encountering undefined or complex numbers, making the solution straightforward.

Square Roots

Square roots are an essential part of many mathematical operations, including simplifications like the one discussed in this exercise. A square root of a number \(x\) is a number \(y\) such that \(y^2 = x\). Here are some key aspects of square roots:

• Finding the square root reverses the operation of squaring, hence \(\sqrt{5}^2\) computation returns \(5\).
• Square roots are frequently involved in the simplifications of expressions, as they naturally balance squaring operations.
• Square roots can produce irrational numbers, but they play a role in attaining precise solutions in real number contexts.

Understanding square roots, such as how they interact in our expression to simplify to rational numbers, equips students with a deeper grasp of their applications in further mathematics. It ties back to the main goal of refinement process—transforming expressions into simple, workable forms easily interpretable and applicable to subsequent arithmetic or algebraic processes.