Problem 42 - Algebra 2

Question

Evaluate each expression. \(\sqrt{9}\)

Step-by-Step Solution

Answer

The value of \(\sqrt{9}\) is 3.

Step 1: Identify the Expression

The given expression to evaluate is \(\sqrt{9}\), which involves finding the square root of 9.

Step 2: Recall the Definition of Square Root

The square root of a number \(x\), represented as \(\sqrt{x}\), is a value \(y\) such that \(y^2 = x\). In other words, \(y\) is the number that, when multiplied by itself, results in \(x\).

Step 3: Calculate the Square Root

Consider possible values for \(\sqrt{9}\). We need to find a number which, when squared (multiplied by itself), equals 9. The number 3 satisfies this condition since \(3 \times 3 = 9\).

Step 4: Verify the Result

Check whether there are any other numbers that squared give 9. Though \(-3\) also squares to 9, the principal square root (usually the positive root) is considered as the default value. Thus, the calculated square root is 3.

Key Concepts

Evaluating Expressions

Evaluating expressions is a foundational skill in mathematics that involves determining the value or solution of a given mathematical statement. An expression like \(\sqrt{9}\) requires finding the principal square root of the number 9. To evaluate such expressions, one typically follows a systematic approach:

• Identify the Expression: Recognize that the task is to find a specific value, in this case, the square root of 9.
• Understand the Mathematical Operation: Square root represents a number that, when multiplied by itself, equals the original number.
• Compute the Answer: Perform the calculation to find the exact value.
• Verify the Solution: Ensure no alternate solutions meet the criteria for the specific instruction (i.e., principal square root).

By following these steps, students can confidently determine values of both simple and complex mathematical expressions.

Mathematical Definitions

Mathematical definitions play a crucial role in understanding and solving problems. For evaluating \(\sqrt{9}\), it is vital to have a clear grasp of what a square root is.
The square root of a number, denoted as \(\sqrt{x}\), is a number \(y\) such that \(y^2 = x\). This means that multiplying \(y\) by itself gives you the number \(x\). For instance, the square root of 9 is 3, because \(3 \times 3 = 9\).
Understanding that the square root function typically refers to the principal (positive) root helps resolve ambiguities, especially when negative numbers might also square to give the original number. For practical purposes, when we write or see \(\sqrt{9}\), we implicitly mean the positive square root, which is 3. This definition ensures consistency and clarity in various mathematical contexts.

Problem Solving Steps

Solving problems involving square roots involves a sequence of logical steps that lead to the correct solution. Let’s outline these steps with \(\sqrt{9}\) as an example.

• Recognize the Problem: Identify that you need to find the square root of the given number, 9.
• Recall Essential Knowledge: Remember definitions and properties related to square roots; knowing that a square root is a value that squares to yield the original number is key.
• Perform Calculations: Test possible numbers to find which, when squared, equals 9. Here, both 3 and -3 are possibilities, since \(3 \times 3 = 9\) and \((-3) \times (-3) = 9\).
• Decide on the Principal Root: Understand that, unless otherwise specified, the principal square root (positive value) is the solution, making the answer 3.
• Conclude and Verify: Confirm there are no other interpretations needed for the context given. For this exercise, the principal root of 9 is indeed 3, solving the problem conclusively.

When solved with these structured steps, math problems become more approachable and errors can be minimized.