Problem 21 - Algebra 1

Question

Simplify the expression. $$2 \sqrt{6}-\sqrt{6}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(2 \sqrt{6}-\sqrt{6}\) is \(\sqrt{6}\).

Step 1 Recognize the Like Terms

Identify the like terms within the expression. In this case, the like terms are \(2 \sqrt{6}\) and \(-\sqrt{6}\).

Step 2 Combine Like Terms

Combine the like terms together. \nWe do this by simply performing the operation present between these terms. Since there is a negative sign between \(2 \sqrt{6}\) and \(\sqrt{6}\), we subtract the two terms to combine them: \(2 \sqrt{6} - \sqrt{6}\). This results in \(\sqrt{6}\).

Key Concepts

Like Terms

In mathematics, like terms are terms that contain the same variables raised to the same power. When working with radicals, like terms will have the same radical part. For example, in the given expression, both terms contain \(\sqrt{6}\) as their radical component.
Understanding like terms is essential because it allows us to simplify expressions by combining them.

• Like terms with radicals: Look for terms with identical root values.
• Numerical coefficients: These are the numbers in front of the radical, which can differ even if the radical part is the same.
• Combining Process: Focuses on combining the numerical coefficients while leaving the radical part unchanged.

Recognizing like terms is your first step toward simplifying radical expressions effectively.

Operations with Radicals

Operations involving radicals follow rules similar to those for regular numbers, but with specific considerations. Radicals often appear in mathematical problems where you perform addition, subtraction, multiplication, or division.

• Adding and subtracting: This requires like terms, specifically the same radical part. You simply add or subtract their coefficients.
• Multiplication: Unlike addition and subtraction, radicals can be multiplied even if their radicands differ.
• Division: Requires rationalizing the denominator, especially if it contains a radical.

It's crucial to respect these operation specificities to ensure accurate results in your computations involving radicals.

Subtracting Radicals

When subtracting radicals, you treat the expression in much the same way as subtracting variables or regular numbers, provided they have the same radical part. The key lies in the similarities of their radical components.
For the expression \(2 \sqrt{6} - \sqrt{6}\), both terms have \(\sqrt{6}\), making it straightforward.

• Align the radicals: Ensure the radicals are the same before subtracting.
• Subtract coefficients: Subtract the numerical coefficients while keeping the radical the same.

In this case, you subtract the coefficients (2 and 1) resulting in just \(1 \sqrt{6}\) or simply \(\sqrt{6}\). This simple method allows you to keep expressions tidy while respecting mathematical conventions.