Problem 55 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ -\sqrt{121} $$

Step-by-Step Solution

Answer

The simplified expression is \(-11\).

Step 1: Identify the Radical Expression

The expression to simplify is \(-\sqrt{121}\). It involves a square root, which we need to simplify.

Step 2: Simplify the Radicand

The radicand is the number inside the square root, which is 121. We need to determine the square root of 121. Since \(121 = 11^2\), the square root of 121 is 11.

Step 3: Apply the Negative Sign

Remember that the original expression is \(-\sqrt{121}\). After simplifying the square root, we have 11. Since there is a negative sign in front of the square root, our final expression becomes \(-11\).

Key Concepts

Understanding Square Roots

Square roots are fundamental concepts in mathematics. They are the opposite of squaring a number. When you take the square root of a number, you are finding a number which, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 times 5 equals 25. When simplifying a square root, the goal is to find this number. In mathematical notation, the square root is represented using the radical symbol \( \sqrt{} \). Understanding how to simplify square roots is key to many areas of math, ensuring ease with problems involving radicals.

What is a Radicand?

The term 'radicand' refers to the number or expression inside a radical symbol. In the expression \(-\sqrt{121}\), the radicand is 121. The radicand is the number you need to simplify when dealing with square roots. In the exercise, the radicand 121 was simplified because we know that 121 is equal to \(11^2\). Thus, the square root of the radicand 121 is 11. Remember, identifying and simplifying the radicand is a crucial step when working with radicals.

Dealing With the Negative Sign

The presence of a negative sign in a radical expression can change the nature of the problem slightly. In the exercise, we have the expression \(-\sqrt{121}\). The negative sign in front of the square root implies that the answer will be negative. After determining that the square root of 121 is 11, we apply this negative sign to the result. Therefore, instead of just 11, the expression becomes \(-11\). It is important to keep track of negative signs in mathematical expressions to avoid errors.

Step-by-Step Solution: A Practical Example

Breaking down problems into clear, manageable steps makes it easier to understand mathematical concepts. Let’s revisit the solution in steps for clarity:

• **Step 1**: Identify the radical expression you are trying to simplify. In our case, it's \(-\sqrt{121}\).
• **Step 2**: Simplify the radicand, which is 121. Since 121 equals \(11^2\), its square root is 11.
• **Step 3**: Apply any remaining elements of the original expression, such as a negative sign. Here, we apply the negative sign, giving us \(-11\).

Following this step-by-step solution ensures that each part of the expression is handled correctly. This methodical approach aids in understanding the process, making sure all steps are clearly executed.