Problem 170 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b} $$

Step-by-Step Solution

Answer

-\ \sqrt{11b}

Step 1: Identify Like Terms

Notice that all terms under the radical are the same: \( \sqrt{11b} \ \). This means they can be combined as like terms.

Step 2: Combine Coefficients

Perform the arithmetic operations on the coefficients of the like terms. This involves adding and subtracting the numbers in front of the radicals: $$ 1 - 5 + 3 $$.

Step 3: Simplify the Coefficients

Compute $$ 1 - 5 + 3 $$: \( 1 - 5 = -4 \), and further \( -4 + 3 = -1 \)

Step 4: Rewrite the Expression

Combine the simplified coefficient with the common radical. Therefore, the expression simplifies to \( -1 \ \sqrt{11b} \). When simplified fully, this can be written as $$ - \ \sqrt{11b} $$.

Key Concepts

like terms

In mathematics, like terms are terms that have the same variables raised to the same powers, regardless of their coefficients. These are important because they can be combined to simplify expressions. For instance, in the exercise, all given terms have the same radical expression \(\text{\}\sqrt{11b}\). Because they share this part, they are considered like terms and can be combined. Identifying like terms helps in simplifying problems by reducing the number of terms, making it easier to solve or understand the expression.

coefficients

Coefficients are the numerical parts of terms that contain variables or radicals. They tell how many times the term’s base element is taken. In the exercise, the coefficients of \(\text{\}\sqrt{11b}\) are 1, -5, and 3. Understanding coefficients is crucial because you perform arithmetic operations directly on them when dealing with like terms. For example, combining the coefficients 1, -5, and 3 helps us simplify the expression. The operations on these coefficients result in the simplified coefficient which is then combined back with the radical expression.

arithmetic operations

Arithmetic operations such as addition, subtraction, multiplication, and division, are basic mathematical processes used to manipulate numbers. In simplifying radical expressions, you'll often use addition and subtraction to combine like terms. In the exercise, we combined the coefficients using these operations. The steps were: \( 1 - 5 + 3 \). First we did 1 - 5, which gives us -4. Then we did -4 + 3, resulting in -1. This final result is then used as the new coefficient for the radical expression.

radicals

Radicals are mathematical symbols that represent the root of a number. The most common radical sign is the square root, written as \(\text{\}\sqrt{}\). Simplifying expressions with radicals involves treating the radicals as like terms if they are similar, as in the exercise where \(\text{\}\sqrt{11b} \) appears multiple times. Each radical term must be simplified separately, but like terms with the same radical can be combined by working with their coefficients, as shown in the provided solution. This method helps in making the expression simpler and easier to handle.