Problem 27 - Algebra 2

Question

Name the property illustrated by each equation. $$ 2 \sqrt{3}+5 \sqrt{3}=(2+5) \sqrt{3} $$

Step-by-Step Solution

Answer

The equation illustrates the distributive property.

Step 1: Identify the Operation

In the given equation, observe that both the left side, \(2 \sqrt{3} + 5 \sqrt{3}\), and the right side, \((2+5) \sqrt{3}\), involve addition of terms. This is essential to identify the mathematical property being used.

Step 2: Recognize the Structure

Notice the structure on both sides of the equation. The left side groups like terms with a common radical, \(\sqrt{3}\), while the right side factors out \(\sqrt{3}\) and adds their coefficients, \(2\) and \(5\).

Step 3: Identify the Property

The property illustrated by changing \(2 \sqrt{3} + 5 \sqrt{3}\) to \((2+5) \sqrt{3}\) is the distributive property. This property states that \(a(b + c) = ab + ac\), applying it in reverse for the current equation.

Key Concepts

Algebraic Operations

Algebraic operations are foundational mathematical procedures that allow us to manipulate expressions and equations to solve for unknowns or simplify expressions. These operations include addition, subtraction, multiplication, and division, and they form the basis of algebraic manipulation.
One of the key features of algebraic operations is their ability to transform complicated expressions into simpler ones. This is achieved through a set of rules and properties, such as the distributive property.
This property allows us to apply multiplication across terms within parentheses:

• Distribution can be seen with expressions like \(a(b + c) = ab + ac\).
• In reverse, it allows us to factor out common terms, as in the example with radicals: \(2\sqrt{3} + 5\sqrt{3} = (2+5)\sqrt{3}\).

Recognizing patterns and applying these operations correctly is essential to transforming and solving algebraic expressions effectively.

Radicals in Algebra

Radicals in algebra involve expressions that include roots, such as square roots or cube roots. These can sometimes make expressions look more complex, but they follow similar mathematical rules as other algebraic terms.
When working with radicals, it's important to understand:

• Like terms with radicals can be combined. This means terms with the same radical part can be added or subtracted, just like numerical coefficients. For example, \(2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}\).
• Radicals can be manipulated using the same algebraic operations as you would with any other terms, following the rules of exponents and properties like distribution.

These principles help in simplifying expressions and solving equations that contain radicals, making problems easier to manage.

Combining Like Terms

Combining like terms is a fundamental process in algebra where terms with the same variables or radicals are simplified by adding or subtracting their coefficients. This reduces the complexity of expressions while maintaining their value.
Here’s a breakdown of combining like terms effectively:

• Identify terms that have exactly the same radical or variable component. In the equation \(2\sqrt{3} + 5\sqrt{3}\), both terms have the radical component \(\sqrt{3}\).
• Add the coefficients of these like terms, which transforms the expression into \( (2 + 5)\sqrt{3} = 7\sqrt{3} \).
• Ensure that only like terms are combined, as mixing different terms can lead to incorrect simplifications.

Mastering this skill is crucial for simplifying and solving algebraic expressions efficiently.