Problem 28 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( 6\sqrt{7} - 14 \)

Step 1: Distribute the First Term

Begin by distributing the first term, which is \( 2\sqrt{7} \), to each term inside the parentheses \((3 - \sqrt{7})\). Multiply \( 2\sqrt{7} \) by 3: \[ 2\sqrt{7} \times 3 = 6\sqrt{7} \]

Step 2: Distribute the Second Term

Now, distribute \( 2\sqrt{7} \) to the second term, \( -\sqrt{7} \): \[ 2\sqrt{7} \times (-\sqrt{7}) = -2(\sqrt{7} \times \sqr0t{7}) = -2 \times 7 = -14 \]

Step 3: Combine Like Terms

Combine the results from Step 1 and Step 2. Adding \( 6\sqrt{7} \) and \(-14\) gives the final expression:\[ 6\sqrt{7} - 14 \]

Key Concepts

Distributive Property

The Distributive Property is a critical concept in algebra that helps simplify expressions. It's used to multiply a single term by each term within parentheses. This property states that for any numbers or variables, \( a(b + c) = ab + ac \). In this exercise, the distributive property allows us to handle complex expressions effectively.

• You start by identifying a term that you'll distribute across the other terms inside the parentheses.
• In our exercise, we distributed \( 2\sqrt{7} \) to both \( 3 \) and \( -\sqrt{7} \).
• This process helps to "unpack" the expression into a simpler form.

Remember, the distributive property is pivotal when dealing with equations involving brackets or parentheses. It simplifies the expression and can help solve equations more efficiently.

Radical Expressions

In mathematics, radical expressions involve roots, such as square roots. The symbol \( \sqrt{} \) represents square roots, which ask "what number squared gives us the original number?"

• In this exercise, \( \sqrt{7} \) is a radical expression.
• When multiplying radicals, it's important to understand that \( \sqrt{a} \times \sqrt{a} = a \).
• For example, \( \sqrt{7} \times \sqrt{7} = 7 \).

Handling radical expressions frequently involves simplifying the radicals. Understanding the multiplication of these expressions can help to reduce and simplify the overall equation. Practice is essential in getting comfortable with these types of mathematical expressions.

Simplification

Simplification in algebra means reducing an expression to its simplest form. From taking apart an expression via the distributive property to combining like terms, simplification makes calculations easier and more understandable.

• Once each term is distributed, look to simplify by combining like terms, if possible.
• In this expression, after distribution, we have two resulting terms: \( 6\sqrt{7} \) and \(-14 \).
• These terms are already in their simplest forms since there are no further like terms to combine.

Simplification is an essential final step, ensuring the expression is as clear and concise as possible. This process helps us to interpret and use algebraic expressions effectively across various problems and applications.