Problem 26 - Intermediate Algebra

Question

Evaluate each square root without using a calculator. See Objective 1 and Example 1. $$ -\sqrt{1} $$

Step-by-Step Solution

Answer

\(-\sqrt{1} = -1\)

Step 1: Identify the Square Root

First, recognize that you need to evaluate the expression \(-\sqrt{1}\). This means you will first find the square root of the number inside the radical, which is 1.

Step 2: Calculate the Square Root

The square root of 1 is 1 because 1 multiplied by 1 equals 1. So, \(\sqrt{1} = 1\).

Step 3: Apply the Negative Sign

Since the original expression is \(-\sqrt{1}\), you need to apply the negative sign to the result from Step 2. Thus, \(-\sqrt{1} = -1\).

Key Concepts

Square Root Evaluation

Square root evaluation is the process of determining a number which, when multiplied by itself, results in the original number under the radical. For example, in the expression \(\sqrt{1}\), you are looking for a number that equals 1 when squared. The key is to find this number, which in simple cases is straightforward. Here, \(\sqrt{1} = 1\) because \(1 \times 1 = 1\). It’s important to remember that every non-negative number has a principal square root, which is always non-negative. This concept forms the basis of evaluating square roots efficiently without using a calculator.

Radical Expressions

Radical expressions are expressions that contain a square root, cube root, or any higher-order root. When dealing with radicals, it’s crucial to simplify them. Simplification involves rewriting the expression in its most basic form.

For example, in the expression \(\sqrt{1}\), there is nothing more to simplify because 1 is already a simple integer. Most often, radicals are simplified by factoring out perfect squares. Remember: the index of the root will determine how you simplify.

Some common steps in simplifying radical expressions include:

• Identifying factors that are perfect squares.
• Breaking down the number into its prime factors.
• Re-writing the radical using the product of its factors.

Understanding radical expressions is vital for simplifying them and solving equations involving them efficiently.

Negative Square Roots

A negative square root indicates that you are taking the negative of a square root value. For a positive number \(a\), \(-\sqrt{a}\) simply means applying a negative sign to the square root of \(a\).

In the exercise, \(-\sqrt{1}\) demonstrates this concept. We first calculated \(\sqrt{1} = 1\). By introducing a negative sign, we alter it to \(-1\). The negative sign is external to the square root operation and does not affect the process of finding the square root itself.

It is essential to keep in mind that while the square root \(\sqrt{1}\) is 1, the expression \(-\sqrt{1}\) modifies it to become \(-1\), illustrating how negative square roots function in expressions.