Problem 84 - College Algebra and Calculus: An Applied Approach

Question

Complete the statement with \(<,=\), or \(\geq$$\sqrt{3}-\sqrt{2} \quad \sqrt{3-2}\)

Step-by-Step Solution

Answer

\(\sqrt{3}-\sqrt{2} < \sqrt{3-2}\)

Step 1: Simplify the First Expression

Evaluate the expression \(\sqrt{3}-\sqrt{2}\). Both of these are known values - approximately 1.73205 and 1.41421, respectively. Therefore the end result is approximately 0.31784.

Step 2: Simplify the Second Expression

Evaluate the expression \(\sqrt{3-2}\), or \(\sqrt{1}\). The square root of 1 is 1.

Step 3: Compare the Two Results

Now that we know \(\sqrt{3}-\sqrt{2} ≈ 0.31784\) and \(\sqrt{3-2} = 1\), we can compare the two. Thus, we see that 0.31784 is less than 1.

Key Concepts

Square Root Simplification

Simplifying square roots is a fundamental part of algebra which allows us to work with root expressions easily. When you see a square root, like \(\sqrt{3}\) or \(\sqrt{2}\), you're looking at the number which, when multiplied by itself, equals the number under the root. For instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

When the number under the square root is not a perfect square (like 2 or 3), we can't simplify the square root to a whole number. Instead, we usually approximate the value using a calculator. The precision of these approximations can vary depending on the context. It's crucial to understand that while simplifying square roots like \(\sqrt{3}\) or \(\sqrt{2}\) may lead to non-integer numbers, each square root has an exact value that is often represented approximately in decimal form.

Numerical Expression Evaluation

When evaluating numerical expressions, you are essentially calculating the value of expressions involving numbers and operations. It's like solving a puzzle where each number and operation is a piece that fits together to reveal the final picture—the value of the expression.

Evaluating expressions requires a good understanding of order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For expressions involving square roots, like \(\sqrt{3}-\sqrt{2}\) or \(\sqrt{3-2}\), we must first simplify the square roots (if possible) and then perform any additional arithmetic operations. In these cases, since both expressions involve square roots that cannot be simplified to integers, we approximate their values and then proceed with the arithmetic operations.

Inequalities

In mathematics, inequalities help us understand the relative size of two values. They are expressions involving less than (\(<\)), greater than (\(>\)), less than or equal to (\(\leq\)), and greater than or equal to (\(\geq\)). When comparing square roots, like in the example of \(\sqrt{3}-\sqrt{2}\) and \(\sqrt{1}\), understanding inequalities allows us to determine which of the two expressions holds a lesser or greater value.

After evaluating each expression, we compare their numerical results. Here, since \(\sqrt{3}-\sqrt{2} \< \sqrt{3-2}\), we identify that the former is less than the latter. Using inequalities, we can capture these relationships symbolically to answer questions and define ranges of solutions in mathematics.