Problem 148 - Elementary Algebra

Question

In the following exercises, simplify. $$ 4 \sqrt{5}+8 \sqrt{5} $$

Step-by-Step Solution

Answer

12 \sqrt{5}

Step 1: Identify like terms

Observe that both terms, \(4 \sqrt{5}\) and \(8 \sqrt{5}\), have the same radical part \( \sqrt{5} \).

Step 2: Combine like terms

Since the radical parts are the same, add the coefficients of the like terms: \( 4 + 8 \).

Step 3: Simplify the coefficients

Add the coefficients: \( 4 + 8 = 12 \).

Step 4: Multiply the simplified coefficient with the radical

Now, multiply the simplified coefficient with the common radical part: \( 12 \sqrt{5} \).

Key Concepts

Like Terms

To simplify radical expressions, the first step is to identify like terms. 'Like terms' in radical expressions mean that the radical part (the part under the square root) is the same. For instance, in the expression \( 4 \,\sqrt{5} + 8 \,\sqrt{5} \), both terms have the same radical part \( \sqrt{5} \). This commonality allows us to group and simplify the expression further. Identifying like terms is crucial for combining terms effectively.

Radical Expressions

Radical expressions involve roots, such as square roots. The term 'radical' comes from the symbol \(\sqrt{}\), which denotes the square root. In an expression like \( 4 \,\sqrt{5} \), the number '4' is the coefficient, and \( \sqrt{5} \) is the radical part. Radical expressions can include numbers (like \( \sqrt{2} \)) or variables (like \( \sqrt{x} \)). Understanding the structure of these expressions helps in simplifying and performing operations on them.

Coefficients

Coefficients are the numerical parts of terms in algebraic expressions. In the term \( 4 \,\sqrt{5} \), '4' is the coefficient. Coefficients tell us how many times the radical part is being counted. When simplifying expressions with like radical parts, you focus on adding or subtracting these coefficients first. For example, with \( 4 \,\sqrt{5} + 8 \,\sqrt{5} \), you would add the coefficients 4 and 8 to get 12.

Addition of Radicals

Addition of radicals requires that the radical parts be identical. Once identified, the coefficients are combined. For example, in \( 4 \,\sqrt{5} + 8 \,\sqrt{5} \), since \( \sqrt{5} \) is the same in both terms, you add the coefficients (4 and 8) to get 12. The final expression is then \( 12 \,\sqrt{5} \). This method of adding the coefficients simplifies the expression, while the radical part remains unchanged. This is similar to combining like variables in algebra.