Problem 36 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ 4 \sqrt{13 x}-6 \sqrt{13 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2 \sqrt{13x}\).

Step 1: Identify Like Terms

In this case, the like terms are \(4 \sqrt{13x}\) and \(-6 \sqrt{13x}\). So, the terms under the square root are the same, which means these terms can be combined.

Step 2: Combine Like Terms

Combine these two terms by addition or subtraction. Here, we're subtracting \(6 \sqrt{13x}\) from \(4 \sqrt{13x}\). So, it's \((4-6) \sqrt{13x}\).

Step 3: Simplify The Expression

After subtracting 6 from 4, we get \(-2 \sqrt{13x}\). So the simplified expression after combining like terms is \(-2 \sqrt{13x}\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental process in algebra. It involves simplifying expressions by merging terms that have the same variable part. In the expression \(4 \sqrt{13x} - 6 \sqrt{13x}\), both terms have the same square root term, \(\sqrt{13x}\). This means they are like terms and can be combined.
To do this:

• Look at the coefficients (the numbers in front) of each term: 4 and -6.
• Perform the necessary arithmetic operation on these coefficients, which is subtraction in this case: \(4 - 6 = -2\).
• The result is multiplied by the common square root, giving us: \(-2 \sqrt{13x}\).

By combining these terms, the expression becomes easier to work with.

Simplifying Algebraic Expressions

Simplifying algebraic expressions helps make expressions more concise and manageable. The goal is to make the expression as simple as possible by performing operations and combining like terms. For example, in \(4 \sqrt{13x} - 6 \sqrt{13x}\), simplifying involves a few straightforward steps.
First, identify parts of the expression that can be combined. Here, it's the terms with the square root. By calculating \(4 - 6\), we simplify the coefficients to get \(-2\). We've effectively reduced the expression to \(-2 \sqrt{13x}\), making it a single, simplified term.
Simplified expressions are not only easier to read but also simpler to use in further calculations.

Understanding Square Roots

Square roots often appear in algebraic expressions and recognizing how to work with them is useful. The square root symbol \(\sqrt{}\) signifies a number that, when multiplied by itself, gives the original number under the root.
In our example, \(\sqrt{13x}\) is the square root term. However, when combining terms like \(4 \sqrt{13x}\) and \(-6 \sqrt{13x}\), the square root part remains unchanged. Only the coefficients, 4 and -6, are affected.
When dealing with square roots:

• Remember that the square root itself does not change as long as the terms under the root are identical.
• Combining only impacts the numerical coefficients outside the square root.

Understanding this helps prevent errors when combining expressions involving square roots, ensuring accurate simplifications.