Problem 9 - Prealgebra

Question

Combine by applying the distributive property. Assume all variables represent positive numbers. $$2 \sqrt{y}+\sqrt{y}+3 \sqrt{y}$$

Step-by-Step Solution

Answer

The simplified expression is \(6\sqrt{y}\).

Step 1: Understanding the Terms

First, identify the similar terms in the expression. The terms given are \(2\sqrt{y}\), \(\sqrt{y}\), and \(3\sqrt{y}\). All these terms contain \(\sqrt{y}\), so they are like terms.

Step 2: Combine Coefficients Using Distributive Property

Use the distributive property to combine the coefficients of the like terms. The distributive property allows us to factor out the common \(\sqrt{y}\). So, we write: \(2\sqrt{y} + \sqrt{y} + 3\sqrt{y} = (2 + 1 + 3)\sqrt{y}\).

Step 3: Simplify the Expression

Calculate the sum of the coefficients: \(2 + 1 + 3 = 6\). Thus, the expression simplifies to \(6\sqrt{y}\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in algebra that allows us to simplify expressions by grouping similar components together. In the given expression:

• All terms involve the radical \(\sqrt{y}\), making them like terms because they share the same variable part.

When you encounter like terms, focus on the coefficients, the numbers in front of the radical or variable. Here, the coefficients are 2, 1 (because \(\sqrt{y}\) is the same as 1\(\sqrt{y}\)), and 3. By adding these coefficients together, you can simplify the expression while keeping the shared radical part unchanged. It’s similar to adding apples and apples together — they remain apples, but in a combined quantity. Thus, \(2\sqrt{y} + \sqrt{y} + 3\sqrt{y} = (2 + 1 + 3)\sqrt{y} = 6\sqrt{y}\). This simplification process not only makes the expression more manageable but also sets the stage for further operations if needed.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. In this exercise, the expression \(\sqrt{y}\) is a square root which falls under the category of radical expressions. Understanding radicals is crucial because:

• They allow us to work with quantities in terms of their roots.
• Often arise in various applications, including geometry and physics.

When handling radicals, you treat the radical part — here, \(\sqrt{y}\) — much like you would treat a variable. It stays consistent across the terms you are combining. This means that when you have expressions like \(2\sqrt{y} + 3\sqrt{y}\), the square root itself is not altered; instead, it's the coefficients you manage and simplify. Understanding this principle leads to clarity in your calculations and helps in transitioning smoothly between algebraic expressions and their simplified forms.

Simplifying Expressions

Simplifying expressions is an important process in algebra and mathematics, allowing us to reduce complex expressions into more manageable forms. It involves several steps, such as combining like terms and using algebraic properties like the distributive property. In our example, simplifying meant:

• Using the distributive property: Factoring out the common term \(\sqrt{y}\).
• Adding the coefficients: \(2 + 1 + 3 = 6\).

These steps show how simplification helps bring clarity to an expression. When you simplify, you clean up any unnecessary parts and consolidate similar ones, making the expression easier to use in computations or further transformations. Simplified expressions are often easier to interpret and solve, hence the emphasis on this skill in any mathematical learning journey. Not only does it aid in efficiency, but it also opens pathways to understanding deeper concepts in mathematics.