Problem 14 - Algebra 2

Question

Find each real-number root. $$ -\sqrt{36} $$

Step-by-Step Solution

Answer

\(-\sqrt{36} = -6\). The real-number root is -6.

Step 1: Understand the Square Root Concept

The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are looking for the square root of 36, which is denoted by \(\sqrt{36}\).

Step 2: Calculate the Positive Square Root

The principal (positive) square root of 36 is 6, because \(6 \times 6 = 36\).

Step 3: Apply the Negative Sign

Since there is a negative sign in front of the square root, the value of \(-\sqrt{36}\) is the negative of the principal square root of 36, which is -6.

Key Concepts

Square Root

When students encounter the term 'square root', it helps to think of it simply as finding a number that produces a given number when multiplied by itself. For example, if we're considering the square root of 36, we look for a number that, when squared, equals 36. This means finding a number 'a' such that a × a = 36. It turns out that 6 is this number because 6 × 6 = 36.

However, it's important to understand that every positive number actually has two square roots: a positive and a negative number. This is because both 6 × 6 and (-6) × (-6) equal 36. Despite this, when we see , the convention is usually to refer to the positive square root.

Negative Square Root

Sometimes, students might be confused on encountering a negative square root, like -. This is simply the negative version of the square root. If the principal square root of 36 is 6, then placing a negative sign in front of the square root symbol indicates we want the negative solution to the square root of 36. Therefore, - is -6. It's a common misunderstanding to perceive this as the 'square root of a negative number', but that's not the case here - we're taking the square root of a positive number, 36, and then applying a negative sign to our positive root result.

Principal Square Root

Now, exploring the term 'principal square root', it's crucial to grasp that this refers to the non-negative square root of a number. This is the value that is typically implied in the absence of any additional notation. So when we see without a negative sign in front, we are talking about the principal square root. For the number 36, the principal square root is 6. This is the value used most often in mathematics unless specifically stated otherwise. It's important for students to recognize that even if a number has both positive and negative roots, the 'principal' term designates the positive one.

Radicals

Dealing with 'radicals' can initially appear intimidating, but it's just another way of referring to the root, most commonly the square root, of a number. The radical symbol, or radix, , is what we use to denote the root. When dealing with the simplification of radicals, students should simplify the number under the radical as much as possible. To effectively work with radicals, one must understand factoring, especially looking for perfect squares within a number. When simplifying, we take out perfect square factors from under the radical, simplifying our expression to its most fundamental form.

As an exercise, when faced with radical expressions, try breaking down the number within the radical to its primary factors – this can make identifying and extracting perfect squares easier, streamlining the simplification process.