Problem 16 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

\(\sqrt{6}\)

Step 1: Recognize all similar types

The expression \(2 \sqrt{6}-\sqrt{6}\) contains two terms of the same type namely, \(2 \sqrt{6}\) and \(-\sqrt{6}\). In algebra, similar types or 'like terms' can be combined.

Step 2: Perform Simple Subtraction

After recognizing the same types, perform simple subtraction on the numerical coefficients \(2-1 = 1\). So, we get the simplified expression as \(\sqrt{6}\).

Key Concepts

Like Terms

When simplifying expressions in algebra, the concept of 'like terms' is crucial. Like terms are terms where variables and their corresponding exponents are identical. This also includes terms under a square root, if they share the same radicand (the number under the square root sign). In the expression given, \(2 \sqrt{6}-\sqrt{6}\), both terms are like terms because they have the same square root, \(\sqrt{6}\).

• This makes it easier to combine them since they have the same underlying mathematical structure.
• Identifying like terms is the first essential step in simplifying expressions.

Recognizing like terms allows us to perform operations like addition or subtraction, consolidating terms to simplify the overall expression.

Subtraction

Subtraction is a basic mathematical operation which is essential in simplifying expressions. In this context, subtraction helps us combine like terms. Considering the expression \(2 \sqrt{6} - \sqrt{6}\), subtraction is applied to the coefficients of these like terms.

• Here, we subtract 1 (implied) from 2.
• This step reduces the two terms into a single, simplified term.

By understanding subtraction, you can simplify expressions quickly and effectively.

Coefficients

Coefficients are the numbers that multiply the variable or the radical part of the term. In \(2 \sqrt{6}\) and \(\sqrt{6}\), the coefficients are 2 and 1 respectively.

• Always note that \(\sqrt{6}\) has an implied coefficient of 1 when no number is present.
• When simplifying expressions with like terms, focus on the coefficients first to combine them.

In our exercise, subtracting the coefficient of 1 from 2 gives us a new coefficient of 1, simplifying the entire expression to \(\sqrt{6}\). Remember, coefficients are key to manipulating algebraic expressions effectively.

Square Roots

Square roots are a type of radical expression and are fundamental in algebra. Here, the term \(\sqrt{6}\) represents the square root of 6. In our exercise, both terms contained this square root, \(2 \sqrt{6} - \sqrt{6}\).

• A square root equation contains radicands that may be multiplied by coefficients.
• Like square roots can be combined or simplified just like variables in algebra.

Understanding how square roots function and their properties allows you to manage and simplify expressions easily. In our example, combining the like terms kept the square root \(\sqrt{6}\) consistent while the coefficients changed. This demonstrates how knowing about square roots contributes to simplifying expressions in algebra.