Problem 59 - Algebra 2

Question

Write a sum of three radicals that contains two like terms. Explain how you would combine the terms. Defend your answer.

Step-by-Step Solution

Answer

Combine \(3\sqrt{2}\) and \(4\sqrt{2}\) to get \(7\sqrt{2}\); final expression: \(7\sqrt{2} + 5\sqrt{3}\).

Step 1: Identify Like Terms

To solve the problem, we need to write a sum of three radicals. Let's choose radicals such as \(3\sqrt{2}\), \(5\sqrt{3}\), and \(4\sqrt{2}\). Here, \(3\sqrt{2}\) and \(4\sqrt{2}\) are like terms because they have the same radical part, \(\sqrt{2}\).

Step 2: Combine Like Terms

To combine the like terms \(3\sqrt{2}\) and \(4\sqrt{2}\), we add their coefficients. This gives us \((3 + 4)\sqrt{2} = 7\sqrt{2}\). The resulting sum is \(7\sqrt{2}\).

Step 3: Write the Combined Expression

After combining the like terms, we write the complete expression: \(7\sqrt{2} + 5\sqrt{3}\). Now, we have a sum of radicals with two terms: one combined term \(7\sqrt{2}\) and one unlike term \(5\sqrt{3}\).

Step 4: Defend the Solution

We combined the like terms \(3\sqrt{2}\) and \(4\sqrt{2}\) because they have the same radical part \(\sqrt{2}\). Adding their coefficients simplifies the expression without changing its value. The term \(5\sqrt{3}\) remains separate since its radical part is different and cannot be combined with \(\sqrt{2}\) terms.

Key Concepts

Like Terms

In algebra, "like terms" are terms that have the same variables raised to the same powers. In the context of radical expressions, like terms have identical radical parts. For example, in the expression \(3\sqrt{2}\) and \(4\sqrt{2}\), both terms contain the radical \(\sqrt{2}\).

This identity in the radical part allows us to combine like terms by simply adding or subtracting their coefficients. The radical part remains unchanged while the coefficients are combined. This is because combining like terms is essentially grouping together the coefficients of identical variables or radicals.

• Example: \(3\sqrt{2}\) and \(4\sqrt{2}\) are like terms because their radicals, \(\sqrt{2}\), match.
• Correct combination results in \((3+4)\sqrt{2} = 7\sqrt{2}\).

If the radicals are different—such as in \(3\sqrt{2}\) and \(5\sqrt{3}\)—these terms are unlike and cannot be directly combined. Understanding this concept helps in simplifying and solving algebraic expressions effectively.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. A common radical sign used is the square root, denoted as \(\sqrt{}\). When dealing with these expressions, it's crucial to identify the "radical part," which is the expression under the root sign.

Radical expressions can often be simplified by factoring and using properties of exponents to break down the numbers under the radical. However, the key point in combining radicals is whether they have the same radical part. Only then can we consider adding or subtracting their coefficients.

• The radical \(\sqrt{2}\) in \(3\sqrt{2}\) indicates a radical expression with a specific root.
• Another example, \(5\sqrt{3}\), involves a different radical part and cannot combine with \(\sqrt{2}\).

Understanding how to manipulate radical expressions is fundamental in solving algebraic problems involving roots. Knowing how to simplify, combine, or separate these expressions is crucial for accurate calculation and interpretation.

Coefficients in Algebra

Coefficients in algebra are the numerical parts of terms that multiply variables or radicals. In an expression like \(3\sqrt{2}\), the number 3 is the coefficient. When dealing with radicals, these coefficients determine how you combine like terms.

When radicals share the same root, you add or subtract their coefficients. Here is a simple way to think about it:

• Identify the radicals: if they match, move to the coefficients.
• Add or subtract coefficients: \(3\sqrt{2} + 4\sqrt{2}\) results in \((3 + 4)\sqrt{2} = 7\sqrt{2}\).
• Different radicals mean coefficients stay separate, e.g., \(3\sqrt{2} + 5\sqrt{3}\).

Through the use of coefficients in simplifying expressions, we can make complex problems much more manageable. By focusing on the numerical values, we maintain the mathematical structure and solve or simplify equations efficiently.