Problem 164 - Elementary Algebra

Question

In the following exercises, simplify. $$ 2 \sqrt{15}+5 \sqrt{15}-9 \sqrt{15} $$

Step-by-Step Solution

Answer

\(-2 \sqrt{15}\)

Step 1: Identify Like Terms

The expression given is composed of terms that all include the square root of 15. These like terms can be combined.

Step 2: Combine the Coefficients

Add and subtract the coefficients of the like terms: \(2 + 5 - 9\).

Step 3: Simplify the Coefficients

Perform the arithmetic operations: \(2 + 5 = 7\) and then \(7 - 9 = -2\).

Step 4: Write the Final Expression

Combine the simplified coefficient with the square root term: \(-2 \sqrt{15}\).

Key Concepts

Like Terms

In algebra, 'like terms' are terms that contain the same variables raised to the same power. This means that only the coefficients of these terms are different. For example, in the expression provided, \(2 \sqrt{15}\), \(5 \sqrt{15}\), and \(-9 \sqrt{15}\) are like terms because they all contain the square root of 15.

To simplify algebraic expressions, identifying like terms is crucial. By combining these terms, you can make complex expressions much simpler.

When combining like terms, you only need to work with the coefficients. The variable part (or in this case, the square root) remains the same.

Coefficients

Coefficients are the numerical part of the terms that multiply the variables or square roots. In the expression \(2 \sqrt{15}\), the coefficient is 2. Coefficients indicate how much of each term you have.

When you have like terms, you can add or subtract their coefficients to combine them into a single term. For the given exercise:

• Add the coefficients: \(2 + 5 = 7\)
• Subtract the next coefficient: \(7 - 9 = -2\)

Now, attach this final coefficient to the common square root term, leading to \(-2 \sqrt{15}\).
This is the simplified form of the given expression.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For instance, the square root of 15 is a value (approximately 3.87) that when squared (multiplied by itself) equals 15.

In the given expression, \(2 \sqrt{15}\), \(5 \sqrt{15}\), and \(-9 \sqrt{15}\) all have the square root of 15. This is why they are like terms. The square root symbol (\sqrt{}) does not change when combining like terms. You only manipulate the coefficients.

Square roots often occur in algebraic problems, so it’s important to understand how to work with them, especially when simplifying expressions.