Problem 169 - Elementary Algebra

Question

In the following exercises, simplify. $$ 3 \sqrt{2 a}-4 \sqrt{2 a}+5 \sqrt{2 a} $$

Step-by-Step Solution

Answer

4 \sqrt{2a} \

Step 1: Identify Like Terms

Notice that all terms in the expression contain the same radical part \( \sqrt{2a} \ \). These can be combined as they are like terms.

Step 2: Combine Coefficients

Add the coefficients of \( \sqrt{2a} \ \). The expression is: \(3 \sqrt{2a} - 4 \sqrt{2a} + 5 \sqrt{2a} \ \). Perform the addition and subtraction of the coefficients: \(3 - 4 + 5 = 4 \ \).

Step 3: Simplify the Expression

After combining the coefficients, the simplified expression is: \(4 \sqrt{2a} \ \).

Key Concepts

Like Terms

When simplifying expressions with radicals, identifying like terms is crucial. Like terms are terms that have the same variables and the same exponents. For radicals, like terms also need to have the same radicand (the number under the radical symbol). In our exercise, each term contains the radical \( \sqrt{2a} \), making them all like terms.

Here's why recognizing like terms is important:

• You can only combine terms that are alike. In other words, \( \sqrt{2a} \) can only be combined with other \( \sqrt{2a} \) terms, not something like \( \sqrt{3a} \) or \( \sqrt{2b} \).
• Combining like terms allows us to simplify expressions, which is a valuable skill in algebra.
• It makes the solving process faster and more accurate.

Before combining terms, always check if they are like terms to avoid errors in simplification.

Combining Coefficients

Once you identify like terms, the next step is to combine their coefficients. Coefficients are the numerical part of the term that is multiplied by the variable or radical part.

For instance, in the expression \( 3 \sqrt{2a} - 4 \sqrt{2a} + 5 \sqrt{2a} \), the coefficients are 3, -4, and 5, respectively. Here's how you combine them:

• Add and subtract the coefficients just like you would with regular numbers.
• Ignore the radical part while combining; you’ll deal with it after combining the coefficients.

Let's perform the combination:

• First, add 3 and -4, which gives -1.
• Then, add -1 and 5, resulting in 4.

So, \( 3 \sqrt{2a} - 4 \sqrt{2a} + 5 \sqrt{2a} = 4 \sqrt{2a} \). Combining coefficients simplifies the process and prepares you for final simplification.

Simplification

The ultimate goal of combining like terms and coefficients is to simplify the expression. Simplification makes it easier to understand and work with mathematical expressions.

After combining the coefficients in the expression \( 3 \sqrt{2a} - 4 \sqrt{2a} + 5 \sqrt{2a} \), we get \( 4 \sqrt{2a} \).

Here's why simplification matters:

• It reduces complex expressions to simpler forms.
• Simplified expressions are easier to use in further calculations and problem-solving.
• Simplified forms can reveal relationships between variables and constants more clearly.

Remember, you can only simplify terms that are alike. By following the steps of identifying like terms and combining their coefficients, you ensure that your final expression is as simple and clear as possible.