Problem 217 - Elementary Algebra

Question

In the following exercises, simplify. $$ 8 \sqrt{13}-4 \sqrt{13}-3 \sqrt{13} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt{13} \ \).

Step 1: Identify like terms

Notice that all the terms have a similar part, \( \sqrt{13} \ \). Identify these like terms so they can be combined.

Step 2: Combine like terms

Add and subtract the coefficients of the \( \sqrt{13} \ \) terms. This gives us \(8 - 4 - 3 \ \sqrt{13} \ \).

Step 3: Simplify the expression

Calculate \(8 - 4 - 3 = 1\). So the simplified form of the expression is \( \sqrt{13} \ \).

Key Concepts

like terms

In algebra, the concept of 'like terms' is essential to simplifying expressions. Like terms are terms that have the same variable part, including the same exponent on those variables. For example, in the expression given, all terms have the similar part \( \sqrt{13} \). This common radical makes them 'like terms'. Unlike terms with different radicals, exponents, or variables, cannot be combined through addition or subtraction. Identifying like terms helps by grouping similar parts, aiding in later steps of combining and simplifying.

combining coefficients

Once we have identified the like terms, the next step is combining their coefficients. The coefficients are the numerical parts of each term. In our example, we have the coefficients 8, -4, and -3. To combine, we simply add or subtract these numbers: \( 8 - 4 - 3 \). This process is the same as basic arithmetic. Combining coefficients consolidates all like terms into one single term, simplifying the overall expression.

simplification

After combining the coefficients, the final step is simplification. Simplification means reducing the expression to its simplest form. From \( 8 - 4 - 3 \sqrt{13} \), calculate the coefficients to get \( 1 \sqrt{13} \). Thus, the expression \( 8 \sqrt{13} - 4 \sqrt{13} - 3 \sqrt{13} \), simplifies to \( \sqrt{13} \). This step ensures the expression is concise and easy to understand, representing the least complicated form of the original expression.