Problem 18 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 3 \sqrt{11}-\sqrt{5}+\sqrt{11} $$

Step-by-Step Solution

Answer

The simplified expression is \(4\sqrt{11} - \sqrt{5}\).

Step 1: Identify Similar Terms

The expression is \(3 \sqrt{11}-\sqrt{5}+\sqrt{11}\). As stated in the analysis, you can only add or subtract similar terms, which are square roots with the same number underneath. Thus, \(3 \sqrt{11}\) and \(\sqrt{11}\) are similar terms because they both involve the square root of 11.

Step 2: Sum the Similar Terms

Add similar terms together. \(3 \sqrt{11}\) and \(\sqrt{11}\) can be added together because they are similar terms. Note that \(\sqrt{11}\) is the same as \(1\sqrt{11}\). So, \(3 \sqrt{11}+1\sqrt{11}=4\sqrt{11}\).

Step 3: Write the Final Expression

Combine the result from Step 2 with the remaining term from the original expression. The remaining term was \(-\sqrt{5}\). Thus, the final expression is \(4\sqrt{11} - \sqrt{5}\).

Key Concepts

Square Roots

The square root is a fundamental mathematical concept. It represents a value that, when multiplied by itself, provides the original number. For example, the square root of 9 is 3, because 3 times 3 is 9. In notation, it's represented as \( \sqrt{\cdot} \). This operation turns into a radical expression when the number isn't a perfect square, such as \( \sqrt{11} \) in our problem.

Understanding square roots includes recognizing that not all square root values are exact whole numbers. For instance, \( \sqrt{11} \) is an irrational number. It cannot be expressed as a simple fraction and its decimal form goes on indefinitely without repeating.

The simplification of square roots involves retrieving any perfect square factors from under the radical. However, in expressions like \( \sqrt{11} \), since 11 is a prime number, it remains as is.

Like Terms

In algebra, like terms refer to terms that have exactly the same variables and powers. When working with radicals, this concept is slightly extended. For radicals, two terms are considered 'like' if they have identical parts inside the square root.

In our example, \(3\sqrt{11}\) and \(\sqrt{11}\) are like terms because they both have the same radicand, which is 11. This is essential for performing arithmetic operations on radicals, as only like terms can be combined through addition or subtraction. Unlike regular algebraic expressions, the coefficients can differ, but the variable parts must match completely.

To simplify, one must sum or subtract the coefficients of these like terms just as you would with algebraic terms. This is what was done when adding \(3\sqrt{11} + 1\sqrt{11}\) to get \(4\sqrt{11}\). Always look for matching radicands to effectively simplify any radical expression.

Addition and Subtraction of Radicals

Adding and subtracting radicals requires careful attention to ensure the terms are like terms. This involves checking whether the radicals have the same number under the square root, often referred to as the radicand.

Consider the expression from our exercise: \(3\sqrt{11} - \sqrt{5} + \sqrt{11}\). You can spot that \(3\sqrt{11}\) and \(\sqrt{11}\) can be combined. On simplifying, it becomes \(4\sqrt{11}\) after adding \(3\) and \(1\) (the coefficient of \(\sqrt{11}\)).

The term \(-\sqrt{5}\) does not combine with any other term since its radicand of 5 is different from 11. Thus, radicals with different radicands must remain separate, much like how different variables in algebraic expressions aren't combined.

• Ensure radicands match to apply operations.
• Add or subtract coefficients as needed.
• Retain different radicals as distinct terms in your final expression.

Learning this helps in simplifying and manipulating expressions with radicals efficiently. Always cross-check to handle radicals correctly and avoid merging unlike terms.