Problem 28 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ 2 \sqrt{7}(3-\sqrt{7}) $$

Step-by-Step Solution

Answer

The simplified expression is \(6 \sqrt{7} - 14\).

Step 1: Distribute the Terms

Use the distributive property to multiply each term inside the parentheses by the term outside. Apply the operation as follows: $$2 \sqrt{7} \times 3 - 2 \sqrt{7} \times \sqrt{7}$$

Step 2: Calculate Each Product

Calculate each multiplication separately: 1. For \(2 \sqrt{7} \times 3\), multiply the coefficients and keep the square root term: $$2 \sqrt{7} \times 3 = 6 \sqrt{7}$$2. For \(2 \sqrt{7} \times \sqrt{7}\), use the fact that \( \sqrt{7} \times \sqrt{7} = 7\): $$2 \sqrt{7} \times \sqrt{7} = 2 \times 7 = 14$$

Step 3: Simplify the Expression

Combine the terms from Step 2:$$6 \sqrt{7} - 14$$The expression is already in its simplest form with no like terms to combine.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions by distributing a factor across terms within parentheses. It allows us to break down more complex expressions and rearrange them to make calculation easier.

This property can be expressed as:

• For any numbers or expressions a, b, and c, we have: \( a(b+c) = ab + ac \)

When using the distributive property, be mindful that each term inside the parentheses is multiplied by the term outside the parentheses. For example, in the expression \(2 \sqrt{7}(3-\sqrt{7})\), each term inside the parentheses (3 and \(-\sqrt{7}\)) is multiplied by \(2\sqrt{7}\). This method helps simplify and solve expressions efficiently, making it a crucial tool in algebraic equations.

Radical Expressions

Radical expressions include roots, like square roots or cube roots. Understanding radicals involves knowing how to simplify them and how they relate to other mathematical operations. For instance, in our exercise, \(\sqrt{7}\) is a radical expression.

Here are some quick tips:

• When multiplying radicals, you can usually combine them if the radicands (the numbers under the square root sign) are the same. For example, \(\sqrt{7} \times \sqrt{7} = 7\).
• Radicals can often be simplified by factoring the radicand into a perfect square. However, \(\sqrt{7}\) is already simplified as 7 is a prime number.

Working with radicals involves recognizing these patterns and applying the appropriate multiplication rules, which helps in simplifying expressions further.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form. In the given exercise, after performing the distributive property and multiplying through the radicals, the expression becomes \(6 \sqrt{7} - 14\).

Here's how you simplify:

• First, apply all multiplication and distribution rules to eliminate parentheses.
• Next, combine all like terms. Like terms are terms that have the same variable raised to the same power, but in our exercise, there are no like terms to combine.

The goal of simplifying is to make the expression as compact and understandable as possible, ensuring no further simplification steps are possible. This makes future calculations easier and more efficient.