Problem 162 - Elementary Algebra

Question

In the following exercises, simplify. $$ 6 \sqrt{5}+3 \sqrt{5}+\sqrt{5} $$

Step-by-Step Solution

Answer

10 \sqrt{5}

Step 1: Combine Like Terms

Identify the like terms in the expression. The terms are all multiples of \( \sqrt{5} \).

Step 2: Add the Coefficients

Add the coefficients of the like terms. The coefficients are 6, 3, and 1 (since \( \sqrt{5} \) has an implied coefficient of 1). So, we have \( 6 + 3 + 1 = 10 \).

Step 3: Rewrite the Expression

Substitute the combined coefficient back in front of the \( \sqrt{5} \). The simplified expression is \( 10 \sqrt{5} \).

Key Concepts

Combining Like Terms

When you simplify expressions involving radicals, it's important to recognize and combine like terms. Like terms are terms that have the same variables, including the same radical part. For example, in the expression \(6 \sqrt{5}+3 \sqrt{5}+\sqrt{5}\), all terms involve \(\sqrt{5}\). This means they are like terms and can be combined. Combining these like terms simplifies the expression and makes it easier to manage. The first step is to ensure all terms you want to combine have the same radical part. If this condition is met, you can go ahead and add or subtract their coefficients.

Coefficients

Coefficients are the numerical parts of terms in an algebraic expression. In the term \(6 \sqrt{5}\), the coefficient is 6. When combining like terms, you only add or subtract their coefficients. For example, in the expression \(6 \sqrt{5}+3 \sqrt{5}+\sqrt{5}\), the coefficients are 6, 3, and 1 respectively (note that \(\sqrt{5}\) has an implied coefficient of 1). By adding the coefficients (6 + 3 + 1), you get 10. Hence, the expression \(6 \sqrt{5}+3 \sqrt{5}+\sqrt{5}\) simplifies to \(10 \sqrt{5}\). This process makes a complex-looking expression clearer and easier to understand.

Square Roots

Square roots are a special kind of radical. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\sqrt{5} \times \sqrt{5} = 5\). Simplifying expressions with square roots often involves recognizing and combining like terms. In our example, each term includes \(\sqrt{5}\), making the process of combination straightforward. Knowing how to manipulate square roots is crucial for solving many algebraic problems. Simplifying them correctly can reveal patterns and solutions that are not immediately obvious. Remember, when working with square roots, always pay attention to the coefficients and ensure the terms you're combining have identical radical parts.