Question

If \(x=\sqrt{25}\), which of the following is true? $$ \begin{aligned} & \text { C } x=5 \\ & x=-5 \\ & x=\pm 5 \\ & \text { I don't know. } \\ & \end{aligned} $$

Short answer

\(x = 5\) is the correct answer.

Step-by-step solution

Define the Problem

The problem enquires about the value of \(x\) when \(x=\sqrt{25}\). The crucial point to show in detail here is calculating the square root of 25.

Calculate Square Root

From the known knowledge, the square root of 25 is either 5 or -5. However, as per definition in mathematics and as is commonly accepted, when a square root is considered with no sign value provided the result is always considered to be positive. As there is no negative sign in front of the square root symbol, the answer is considered to be positive. Therefore, \(x = 5\).

Answer the Question

Among the multiple choices provided in the question, \(x = 5\) is the correct statement. Other statements are untrue based on conventional mathematics principles. The statement \(x = -5\) is incorrect because the square root of a number is always positive in the real number system. The statement \(x = \pm 5\) is incorrect because the square root is positive when no sign is mentioned. Therefore, the answer is option C, \(x = 5\).

Key concepts

Mathematical Definitions

Understanding basic mathematical definitions is crucial for solving problems related to square roots and other mathematical operations. A square root is essentially a value that, when multiplied by itself, gives the original number. For example, when we talk about the square root of 25, we are referring to the number that, when multiplied by itself, results in 25. The typical symbol used for the square root is \( \sqrt{} \), and when using this without a negative sign, we imply the principal square root, which is the positive value.

• Principal Square Root: The positive square root of a non-negative number.
• Radicand: The number under the square root symbol.
• Square: A number multiplied by itself, such as 5 becomes 25.

When dealing with square roots, it's important to use the correct mathematical definitions to correctly interpret the results and apply them to solve problems effectively.

Positive Numbers

When it comes to understanding square roots, it's vital to grasp the concept of positive numbers. A positive number is any number greater than zero. Positive numbers are generally considered when discussing the solutions to square roots when no signs are indicated. In everyday math problems, when you see \( \sqrt{} \), and there is no negative sign before it, you are dealing with only the positive square root.For instance, with \( x = \sqrt{25} \), the solution you're probably looking for is 5 because there is no negative attached to the square root symbol. This aligns with the convention that square roots typically refer to the principal (positive) root unless otherwise stated. Understanding these foundations helps you navigate both simple and complex math problems involving square roots and other calculations.

Mathematics Principles

The foundational principles of mathematics guide us to consistent and correct answers. Understanding these principles helps clarify why the square root of a number is defined as positive by default. In mathematics, when we use the square root symbol \( \sqrt{} \), it is understood to represent the positive aspect of the number unless a negative sign manually precedes it.This principle is reflected in the problem-solving process. Here are some key points about this principle:

• By default, \( \sqrt{} \) denotes the principal (positive) square root.
• Mathematical conventions help reduce ambiguity by providing consistency across different problems.
• Negative roots must be explicitly stated, as the principal root is always assumed to be positive.

Applying these principles, particularly when calculating square roots, ensures that our answers are not only correct but also aligned with standard mathematical practices. This uniformity helps us communicate clearly and solve problems more effectively.