Problem 173 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form is \(4 \sqrt{3ab} \).

Step 1 - Identify Like Terms

In the expression given, identify the like terms. The terms are:\(5 \sqrt{3ab}\), \(\sqrt{3ab}\), and \(-2 \sqrt{3ab}\).

Step 2 - Combine Like Terms

Combine the coefficients of the like terms. The combination is:\(5 \sqrt{3ab} + 1 \sqrt{3ab} - 2 \sqrt{3ab}\).

Step 3 - Simplify the Coefficients

Add and subtract the coefficients:\(5 + 1 - 2 = 4\).So, the simplified form is:\(4 \sqrt{3ab}\).

Key Concepts

like terms

Like terms are terms that contain the same variables raised to the same power. In algebra, recognizing like terms is crucial.
This ensures that we only combine terms that can legally be added or subtracted.
For example, in the expression given, the terms are:

• \(5 \sqrt{3ab}\)
• \(\sqrt{3ab}\)
• \(-2 \sqrt{3ab}\)

All of these terms include the expression \(\sqrt{3ab}\). That makes them like terms and can be combined.

combining coefficients

Once you have identified the like terms, you combine their coefficients. Coefficients are simply the numbers in front of a term. For instance:

• In \(5 \sqrt{3ab}\), the coefficient is 5.
• In \(\sqrt{3ab}\), the coefficient is 1 (since a single term without a number always implies 1).
• In \(-2 \sqrt{3ab}\), the coefficient is -2.

These coefficients can be added or subtracted from each other while keeping the square root term \(\sqrt{3ab}\) as it is. This results in:
\(5 + 1 - 2\).
Simplifying this, you get 4. Thus, combining the coefficients gives you the term \(4 \sqrt{3ab}\).

simplification steps

To simplify an algebraic expression effectively:

• Step 1: Identify the like terms. This involves recognizing terms that have the same root expression or variable combination.
• Step 2: Combine the coefficients. This means adding or subtracting the numerical parts of the like terms.
  In our example, you performed:\(5 + 1 - 2 = 4\).
• Step 3: Write the simplified expression. After combining the coefficients, place this number in front of the common square root or variable term.

So, the simplified form for the given example is:\(4 \sqrt{3ab}\).Remember, simplifying expressions helps make solving equations easier and understanding relationships between terms clearer.