Problem 47 - Algebra 2

Question

Evaluate each expression. \(-\sqrt{25}\)

Step-by-Step Solution

Answer

The evaluated expression is -5.

Step 1: Evaluate the Square Root

Find the square root of the number inside the radical. The expression inside the square root is 25. Since 25 is a perfect square, its square root is 5. Therefore, \( \sqrt{25} = 5 \).

Step 2: Apply the Negative Sign

Now apply the negative sign in front of the square root. We need to multiply the square root result by -1. Therefore, \( -\sqrt{25} = -1 \times 5 = -5 \).

Key Concepts

Understanding Perfect Squares

When we talk about perfect squares, we are discussing numbers that result from multiplying an integer by itself. A perfect square is always a non-negative whole number. This means if you take a number like 5 and multiply it by itself, you get 25, which is a perfect square. Thus, saying "25 is a perfect square" simply means there exists an integer whose square is 25.

Common examples of perfect squares include:

• 1 (since 1 x 1 = 1)
• 4 (since 2 x 2 = 4)
• 9 (since 3 x 3 = 9)
• 16 (since 4 x 4 = 16)
• 25 (since 5 x 5 = 25)

Recognizing perfect squares helps in evaluating square roots quickly and accurately. When you know a number is a perfect square, finding the square root is simply identifying which number was squared to get there.

Evaluating Expressions Involving Square Roots

Evaluating expressions involves several steps, ensuring you simplify or solve math problems correctly. When we evaluate an expression like \(-\sqrt{25}\), a typical approach is to break it into the tasks of finding the square root first and then addressing any additional operations such as addition, subtraction, or multiplication.

For \(-\sqrt{25}\):

• Step 1: Recognize that 25 is a perfect square, and its square root is 5.
• Step 2: Assess any symbols or operations outside the square root. In this case, there is a negative sign in front, signifying multiplication by -1.

Therefore, you would calculate the square root first to get 5, then multiply it by -1, resulting in -5. Breaking down expressions into these manageable steps helps ensure accuracy in evaluating more complex expressions.

Working with Negative Numbers

Negative numbers can seem tricky at first, but they're an essential part of math. In the expression \(-\sqrt{25}\), the negative sign indicates that the final result should be less than zero. Applying the negative sign means you take the positive result from the square root and make it negative.

Here's how it works:

• The square root of 25 is 5, a positive number.
• The negative sign in front represents multiplying the result by -1.
• Thus, \(-\sqrt{25} = -1 \times 5 = -5\).

Understanding negative numbers is crucial because they appear in various areas of mathematics, from simple operations to advanced equations. They allow us to represent values below zero, which is important in many practical contexts like temperature, banking, and physics.