Problem 152 - Elementary Algebra

Question

In the following exercises, simplify. $$ 9 \sqrt{n}+3 \sqrt{n} $$

Step-by-Step Solution

Answer

The simplified expression is \[12 \sqrt{n}\].

Step 1 - Identify Like Terms

In the given expression, identify the terms that are like terms. In this case, both terms are like terms because they both contain the square root of n, written as \(9 \sqrt{n}\) and \(3 \sqrt{n}\).

Step 2 - Combine Like Terms

Add the coefficients of the like terms together. The coefficients are 9 and 3 respectively. So, you perform the operation: \[9 + 3 = 12\].

Step 3 - Simplify the Expression

After combining the coefficients, multiply the sum by the common factor \sqrt{n}\. Hence, the simplified expression becomes \[12 \sqrt{n}\].

Key Concepts

Like Terms

In algebra, identifying and working with 'like terms' is an essential skill. 'Like terms' are terms that have the same variable part. For example, in the expression \(9 \sqrt{n} + 3 \sqrt{n}\), both terms are like terms because they both contain the variable \( \sqrt{n} \). Identifying like terms makes it easier to simplify an algebraic expression.
Remember, like terms must have the exact same variable parts—including powers and roots—not just similar or related terms.

Coefficients

Coefficients are the numerical parts of algebraic terms. In the expression \(9 \sqrt{n} + 3 \sqrt{n}\), the numbers 9 and 3 are the coefficients. These coefficients tell you how many times the variable part of the term is being multiplied.
When you combine like terms, you add or subtract their coefficients while keeping the variable part unchanged. For instance, adding the coefficients 9 and 3 from the example gives you 12.

Square Roots

Square roots appear frequently in algebraic expressions. The square root of a number \( n \) is a value that, when multiplied by itself, will give \( n \). In the example \(9 \sqrt{n} + 3 \sqrt{n}\), \( \sqrt{n} \) is the square root of some variable \( n \).
Square roots are considered in expressions the same way as any other variable part. When you simplify terms involving square roots, identify any like terms and proceed to combine their coefficients.

Simplification

Simplification involves making an algebraic expression as short and clean as possible. For the expression \(9 \sqrt{n} + 3 \sqrt{n}\), you start with identifying like terms and their coefficients. Once identified, you add the coefficients: \[9 + 3 = 12\].
Finally, you multiply the sum of the coefficients by the common variable part: \[12 \cdot \sqrt{n} = 12 \sqrt{n}\].
This process not only shortens your expression but also makes it easier to work with in further calculations or solve in equations.