Problem 62 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ -\sqrt{400} $$

Step-by-Step Solution

Answer

-20

Step 1: Identifying the Square Root

Identify the square root of 400. The square root of 400 is \( \sqrt{400} = 20 \) because when 20 is multiplied by itself ( \( 20 \times 20 \) ), we get 400.

Step 2: Applying the Negative Sign

Apply the negative sign to the square root. This gives us \( -\sqrt{400} = -20 \).

Key Concepts

Evaluating Expressions

Evaluating an expression means breaking it down and processing it to find its value. When faced with an expression like \(-\sqrt{400}\), it's important to methodically determine what each part represents and calculates.

In our case, looking at \(\sqrt{400}\) means finding which number can square itself to give 400. Once identified, the expression's value can be calculated by applying the appropriate signs and operations.

• First, calculate the square root, identifying the number that, when multiplied by itself, equals the value under the root symbol.
• Second, apply any additional operations, such as multiplying by -1 to incorporate a negative sign.

Understanding these steps allows you to accurately evaluate and solve more complex expressions later.

Exact Value

The exact value in mathematics refers to the precise and unrounded result of an evaluation. For example, the square root of 400 is exactly 20, as 20 multiplied by itself equals 400.

Obtaining the exact value is crucial whenever possible, as it provides the most accurate result. It allows mathematicians and students to work with precision, which is necessary for solving problems correctly.

• To find the exact value of a square root, consider perfect squares, which are numbers like 400, 100, or 144, that have whole numbers as their roots.
• The exact value doesn't require approximation, making it preferable over rounded numbers.

Knowing when an exact value is required versus when approximation is allowed saves time and reduces errors.

Negative Sign

A negative sign in mathematics can change the scale and direction of a number's value. In the expression \(-\sqrt{400}\), the negative sign indicates that instead of a positive outcome, the result will be the negative version of the square root.

Handling negative signs correctly is essential to accurately solving expressions. Missing or misapplying a negative sign can lead to incorrect conclusions.

• Place a negative sign in front of a root or number alters the final answer by making it a negative value.
• It's crucial to apply the negative sign to the expression after evaluating any other mathematical operations like finding square roots.

By consistently checking for negative signs, you can ensure accurate calculations and outcomes.

Mathematical Operations

Mathematical operations are the cornerstone of problem-solving in math. They include actions like addition, subtraction, multiplication, and division. Each operation has a specific function and order when evaluating expressions.

For the expression \(-\sqrt{400}\), the main operation is finding the square root, followed by multiplication with the negative sign. These steps must be carried out systematically.

• Understand the order of operations to solve expressions correctly. Use techniques like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to guide this process.
• Always perform operations inside parentheses first, and simplify complex expressions step by step.

By mastering these operations, you can confidently approach various math problems and apply suitable solutions strategically.