Problem 74 - Algebra 2

Question

Simplify. $$ 5 \sqrt{3}-4 \sqrt{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt{3} \).

Step 1: Identify Like Terms

Notice that both terms include the square root of 3, making them like terms. In algebraic expressions, like terms have the same variable and exponent combination - in this case, it's the root term \( \sqrt{3} \).

Step 2: Subtract the Coefficients

Since \( 5\sqrt{3} \) and \( 4\sqrt{3} \) have the same radical term, you can subtract their coefficients. Subtract 4 from 5: \( 5 - 4 = 1 \)

Step 3: Combine the Simplified Terms

After subtracting the coefficients, combine them with the common radical, \( \sqrt{3} \). This results in: \( 1\sqrt{3} \), which is simply \( \sqrt{3} \).

Key Concepts

Like Terms

In algebra, **like terms** are terms that contain the same variables raised to the same power. They can be easily combined. This is crucial when simplifying expressions as it allows us to perform operations like addition and subtraction.

### Example with Square Roots
In the expression \(5 \sqrt{3} - 4 \sqrt{3}\), both terms have the same radical component, \(\sqrt{3}\), which makes them like terms. You can spot like terms by comparing their radicals; they must be identical for the terms to be considered alike.

**Why focus on Like Terms?**
- Simplify complex expressions by grouping similar terms
- Reduce expressions to a more manageable form

By identifying and combining like terms, you're essentially "cleaning up" the expression, making it easier to evaluate or further simplify.

Coefficients

**Coefficients** are the numerical parts of terms that usually stand in front of variables or radicals. They are crucial in operations like addition and subtraction when dealing with like terms.

### Using Coefficients in Simplification
In the expression \(5 \sqrt{3} - 4 \sqrt{3}\):

• The coefficients are \(5\) and \(4\).
• These numbers tell you how many times you are counting the radical \(\sqrt{3}\).
• To combine the terms, subtract the coefficients: \(5 - 4 = 1\).

This operation shows you how the terms can be simplified while keeping the radical part unchanged. Numbers (coefficients) take lead roles in calculations, directing how like terms are merged or simplified.

Radicals

A **radical** refers to the symbol \(\sqrt{}\) used to indicate the root of a number. Radicals are powerful tools in mathematics that help solve equations and simplify numbers to their roots.

### Simplifying Radicals
In the example \(5 \sqrt{3} - 4 \sqrt{3}\):

• The radical here is \(\sqrt{3}\), a square root which stays constant when simplifying like terms.
• Radicals allow us to keep expressions exact rather than approximating decimal values.

This lets us maintain precision in mathematical expressions. It’s crucial while combining like terms that involve radicals, as you maintain the radical part unchanged and focus on the coefficients to execute the simplification.