Problem 20 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$\sqrt{25}-\sqrt{4}$$

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Evaluate the Square Root of 25

To start simplifying the expression \( \sqrt{25} - \sqrt{4} \), we need to find the square root of each number. The square root of 25 is 5 since \(5 \times 5 = 25\).

Step 2: Evaluate the Square Root of 4

Next, we calculate the square root of 4. The square root of 4 is 2 because \(2 \times 2 = 4\).

Step 3: Substitute and Simplify the Expression

Now, replace \(\sqrt{25}\) with 5 and \(\sqrt{4}\) with 2 in the original expression. This results in \(5 - 2\). Then, subtract 2 from 5 to get 3.

Key Concepts

Understanding Square Roots

Square roots are an essential part of mathematics. They allow us to determine what number, when multiplied by itself, will equal a given number. For instance, the square root of 25, represented as \( \sqrt{25} \), is 5. This is because \( 5 \times 5 = 25 \). Similarly, \( \sqrt{4} \) is 2, since \( 2 \times 2 = 4 \).

• Perfect Squares: Numbers like 25 and 4 are known as perfect squares because they are the squares of whole numbers.
• Notation: The square root symbol \( \sqrt{} \) is used to denote the root and indicates a reverse operation to squaring.

Understanding square roots simplifies complex mathematical operations by breaking them down into simpler, more manageable components. This ability is crucial for solving various algebraic expressions.

Basic Arithmetic for Simplifying Expressions

Basic arithmetic forms the foundation of simplifying mathematical expressions. To simplify \( \sqrt{25} - \sqrt{4} \), we start by converting square roots into whole numbers, a process we've already defined through square roots.Once each root is evaluated:

• The square root of 25 equals 5.
• The square root of 4 equals 2.

The arithmetic operation remaining is subtraction: \( 5 - 2 \). This simple step involves taking 2 away from 5, which results in 3. Mastery of basic arithmetic operations like addition, subtraction, multiplication, and division is crucial in evaluating and simplifying mathematical expressions.

Working with Mathematical Expressions

A mathematical expression consists of numbers, operators, and sometimes variables, set into a meaningful arrangement representing a specific value or concept. In our example, \( \sqrt{25} - \sqrt{4} \) is an expression involving square root operations and subtraction.

• Simplification: Simplifying expressions often involves operations such as replacing square roots with whole numbers, performing arithmetic calculations, and reducing the expression to its simplest form.
• Order of Operations: Following the correct order of operations is essential. With expressions like \( a - b \), you directly perform the operation once parts a and b are simplified.

Utilizing principles of simplification and understanding the impact of each component and operation in an expression ensures accurate and clear mathematical interpretations.