Problem 2 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(7 \sqrt{12}+4 \sqrt{3}\)

Step-by-Step Solution

Answer

The simplified form is \(18\sqrt{3}\).

Step 1: Identify Terms to Distribute

The expression given is \(7\sqrt{12} + 4\sqrt{3}\). We can simplify by distributing, which means breaking down \(\sqrt{12}\) into its simplest radical form.

Step 2: Simplifying the Radicals

Recognize that \(\sqrt{12} = \sqrt{4 \times 3}\). Using the property of radicals, \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\), this becomes \(\sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). Replace \(\sqrt{12}\) in the expression with \(2\sqrt{3}\).

Step 3: Substitute and Distribute

Substitute \(2\sqrt{3}\) for \(\sqrt{12}\) in the original expression, resulting in \(7(2\sqrt{3}) + 4\sqrt{3}\). Now apply the distributive property: \(7 \times 2\sqrt{3} = 14\sqrt{3}\).

Step 4: Combine Like Terms

Now add the like terms: \(14\sqrt{3} + 4\sqrt{3}\). Since both terms have \(\sqrt{3}\) as a common factor, combine them: \(14 + 4 = 18\). Therefore, the expression simplifies to \(18\sqrt{3}\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down a complex expression into its simplest form. This can often make calculations more manageable and help reveal the underlying structure of the expression. Here, the key step was to simplify the radical part of the expression. By rewriting \(\sqrt{12}\) as \(2\sqrt{3}\), it becomes easier to work with. This process is essential because a simplified expression is not only aesthetically pleasing but also ensures easier manipulation in further calculations.

• Start by identifying parts of the expression that can be simplified.
• Break down complex terms into simpler components by finding factors or using mathematical properties.
• Substitute the simplified components back into the original expression.

Breaking expressions down into their simplest forms is a foundational skill in algebra, allowing for easier comparison and manipulation of algebraic terms.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. They can often be intimidating because they involve non-integer values, but mastering them is all about understanding how to simplify these expressions.

• This often involves breaking down the radicand (the number under the radical sign) into its prime factors.
• Find perfect squares within those factors when working with square roots. For example, \(\sqrt{12}\) can be rewritten as \(\sqrt{4 \times 3}\), which further simplifies to \(2\sqrt{3}\).
• Recognize that different radical expressions can sometimes be combined, but only if they share the same radicand after simplification.

Radical expressions demand careful attention to detail, yet they follow consistent rules that, once understood, make them much less daunting.

Combining Like Terms

Combining like terms is a crucial process in algebra that makes expressions more manageable by merging terms with the same variables or radicals. In the provided exercise, terms like \(14\sqrt{3}\) and \(4\sqrt{3}\) were combined because they both contain the radical \(\sqrt{3}\).

• Identify coefficients and constant terms that multiply by the same variable or radical.
• Sum their coefficients while keeping the common factor (like the radical part) unchanged.
• This step is similar to combining similar items for simplification, making further algebraic procedures straightforward.

Overall, combining like terms not only helps to simplify expressions but also prepares them for any potential algebraic manipulations or function evaluations that follow.