Problem 2 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(1-12,\) or indicate that the root is not a real number. $$\sqrt{25}$$

Step-by-Step Solution

Answer

The square root of 25 is \(5\).

Step 1: Understanding Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are asked to find the number that, when multiplied by itself, gives 25.

Step 2: Calculating Square Root

The square root of 25 is either \(5\) or \(-5\) because \(5*5 = 25\) and \(-5*(-5) = 25\). But, commonly, when we refer to the square root of a positive real number, we usually refer to the positive root only.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers that can be found on the number line. This encompasses all rational numbers, like fractions and decimals, as well as irrational numbers that cannot be expressed as simple fractions, like \( \sqrt{2} \\) and \( \pi \\). Real numbers can be both positive and negative, and they also include zero.

• Rational Numbers: Numbers that can be expressed as the quotient of two integers (e.g., 1/2, 4, -3).
• Irrational Numbers: Numbers that cannot be expressed as a quotient of two integers. Their decimal expansions are non-repeating and non-terminating.
• Whole Numbers and Integers: Whole numbers are non-negative numbers including zero, and integers are whole numbers that can be positive or negative.

Real numbers are essential because they allow us to quantify and make sense of the world around us, and they serve as the building block for more advanced mathematical concepts.

Evaluating Expressions

Evaluating expressions involves finding the value of the expression when you substitute the variables or perform mathematical operations. In this context, we focus on finding the square root, which is a common operation.
When presented with the expression \( \sqrt{25} \\), we are looking for a number that, when squared, gives 25. Simplifying, we find both \( 5 \\) and \( -5 \\) because:

• \( 5 \times 5 = 25 \)
• \( -5 \times -5 = 25 \)

However, in typical practice, when we solve for \( \sqrt{n} \\) for positive \(n\\), we take the principal square root—this is the non-negative root. The principal root is often the one used in real-world calculations and broader mathematical contexts.

Positive and Negative Roots

Positive and negative roots arise from the properties of square numbers. Since squaring a number involves multiplying it by itself, both a positive number and its negative counterpart can produce the same result. Thus, both \( 5 \\) and \( -5 \\) are roots for \(\sqrt{25}\\), because:

• \( 5 \times 5 = 25 \)
• \( -5 \times -5 = 25 \)

When solving problems, it's important to know the context.
In many cases, such as in calculus or practical applications, we use the positive root, known as the principal root. This is denoted by the radical symbol without a negative sign—that is, \( \sqrt{25} = 5 \\).
Understanding which root to use is crucial when solving real-world problems, as they may yield very different outcomes.