Problem 34 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. \(8 \sqrt{5}+11 \sqrt{5}\)

Step-by-Step Solution

Answer

The simplified expression is \(19 \sqrt{5}\)

Step 1: Identify Similar Terms

The given equation is \(8 \sqrt{5}+11 \sqrt{5}\). Both terms in this equation are multiples of \(\sqrt{5}\). Therefore, they are similar terms and can be added together.

Step 2: Combine Similar Terms

In this step, combine the similar terms together. To do so, add the coefficients of the similar terms: \(8+11 = 19\). This gives us the simplified expression, which is \(19 \sqrt{5}\)

Key Concepts

Similar Terms

In algebra, identifying similar terms is a fundamental skill—especially when simplifying expressions involving radicals. Similar terms, sometimes known as "like terms," are terms whose variables (and their exponents) are precisely the same. When it comes to radicals, similar terms are those that have the same radical part. This means you're essentially looking at the number inside the square root, regardless of the coefficients that appear in front.

• Radical Part: For example, in the expression \(8\sqrt{5} + 11\sqrt{5}\), both terms have the same radical part, \(\sqrt{5}\).
• Simplifying Radicals: When simplifying, it’s crucial to pinpoint these similar terms first, as they need the same radical so they can be handled together.

Knowing to look for and combine similar terms forms a solid basis in streamlining expressions and is especially helpful for solving equations efficiently.

Coefficients

Understanding coefficients is essential when working with algebraic expressions. A coefficient is the numerical factor that is multiplied by the variable or radical part of a term. In any given term of the form \(a\sqrt{b}\), \(a\) stands for the coefficient. It tells you how many times the radical is being taken.

• Reading Coefficients: In \(8\sqrt{5}\), the coefficient is \(8\), and in \(11\sqrt{5}\), it’s \(11\).
• Adding Coefficients: When adding similar radical terms, the coefficients are summed, leaving the radical part unchanged. This is a critical step in simplifying radical expressions.

Being comfortable with coefficients allows you to execute addition or subtraction of radicals effectively, making problem-solving much more manageable.

Addition of Radicals

Adding radicals might seem tricky, but it becomes intuitive with practice. When adding radicals, focus on the coefficients of similar terms, just like you would with regular algebraic terms. The radical part remains unchanged during this process.

• Combine Coefficients: In our given problem, \((8\sqrt{5} + 11\sqrt{5})\), we add the coefficients directly: \(8 + 11 = 19\).
• Preserve the Radical: The radical part \(\sqrt{5}\) should stay as it is, resulting in the expression being \(19\sqrt{5}\).
• Practical Application: This technique is useful not only for simplifying expressions but also when solving equations involving radicals in algebra and beyond.

This approach to adding radicals allows for a simplified expression and ensures accuracy in larger computations.