Problem 296 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -\sqrt{121} $$

Step-by-Step Solution

Answer

-11

Step 1: Understand the square root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 121 is a number that, when multiplied by itself, equals 121.

Step 2: Identify the square root

Determine the principal (positive) square root of 121. We know that \(11 \times 11 = 121\), so \(\sqrt{121} = 11\).

Step 3: Apply the negative sign

The given expression includes a negative sign outside the square root. Therefore, \( -\sqrt{121} = -11 \).

Key Concepts

Square Root

A square root is a fundamental concept in mathematics. It is a number that produces a specified quantity when multiplied by itself. For instance, the square root of 121 is found by figuring out what number, when squared (i.e., multiplied by itself), equals 121.
To break it down:

• When we say the 'square root of 121', we are looking for a number x such that \[ x^2 = 121 \].
• In this problem, we find that \[11 \times 11 = 121 \], so \[ \text{the square root of } 121 \text{ is } 11 \].

Understanding square roots is crucial for simplifying many mathematical expressions and solving equations.
Think of the square root as the inverse operation to squaring a number.
This makes it easier to comprehend.

Negative Sign

Negative signs in mathematics are used to indicate that a number is less than zero. In the context of square roots, a negative sign before the square root symbol alters the value's sign.
Here’s how it works:

• If you see \( -\frac {square root of}{S} \), it means you are taking the positive square root of S and then multiplying it by -1.
• For example, \( -\frac{square root of}{121} = -11 \).

The negative sign applies directly to the result of the square root, not to the value under the square root.
It’s similar to normal multiplication and division rules but with square root operations.

Principal Square Root

The principal square root is the positive square root of a number.
When we talk about square roots, there are usually two solutions: one positive and one negative.
The principal square root refers specifically to the positive one.
For example:

• The number 121 has two square roots: +11 and -11.
• However, \(\frac{square root of}{121} = 11\),the principal square root is 11.

This designation helps avoid confusion when dealing with positive roots in mathematical operations.
Keep in mind that the principal square root is frequently implied when no negative signs are present.
It’s crucial to remember this concept for correctly simplifying expressions.