Problem 21 - Intermediate Algebra

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ -4 \sqrt{5}(2 \sqrt{5}+4 \sqrt{12}) $$

Step-by-Step Solution

Answer

The product is \(-40 - 32 \sqrt{15}\).

Step 1: Distribute the Term

Begin by distributing \(-4 \sqrt{5}\) into each term inside the parentheses: \((2 \sqrt{5} + 4 \sqrt{12})\). This results in two separate expressions: \((-4 \sqrt{5}) \cdot (2 \sqrt{5})\) and \((-4 \sqrt{5}) \cdot (4 \sqrt{12})\).

Step 2: Multiply First Pair of Terms

For the first expression, multiply \(-4 \sqrt{5} \cdot 2 \sqrt{5}\):\[-4 \sqrt{5} \cdot 2 \sqrt{5} = -8 (\sqrt{5} \cdot \sqrt{5}) = -8 \times 5 = -40.\]

Step 3: Simplify Second Pair of Terms

For the second expression, first simplify \(\sqrt{12}\). Given \(\sqrt{12} = \sqrt{4 \times 3} = 2 \sqrt{3}\), substitute back into the original expression. Multiply:\[-4 \sqrt{5} \cdot 4 (2 \sqrt{3}) = -16 \times 2 \sqrt{15} = -32 \sqrt{15}.\]

Step 4: Combine the Results

Combine the simplified results from both the multiplications: \(-40 + (-32 \sqrt{15})\). Therefore, the final expression is:\[-40 - 32 \sqrt{15}.\]

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental principle in algebra used to multiply a single term with two or more terms inside parentheses. It states that for any real numbers, the expression \(a(b + c)\) is equal to \(ab + ac\). In our exercise, we apply this property to the expression

• \(-4 \sqrt{5}(2 \sqrt{5} + 4 \sqrt{12})\)

This meant distributing \(-4 \sqrt{5}\) to both terms within the parentheses, resulting in two separate products:

• \((-4 \sqrt{5}) \cdot (2 \sqrt{5})\)
• \((-4 \sqrt{5}) \cdot (4 \sqrt{12})\)

By doing this, we simplify the original expression into parts that are much easier to manage and solve further.

Simplifying Radicals Step-by-Step

Simplifying radicals is the process of transforming a radical expression into its simplest form. A radical expression is a number under a square root symbol, such as \(\sqrt{12}\). To simplify \(\sqrt{12}\), first identify and factor its components:

• \(\sqrt{12} = \sqrt{4 \times 3}\)

Recognizing that \(4\) is a perfect square, we can simplify it to:

• \(2 \sqrt{3}\)

Returning to the exercise, we use this substitution in our second product, making the multiplication easier. The process of simplifying radicals can make long expressions shorter and more intuitive. This is crucial for clear calculations, especially with additional operations like multiplication and addition.

Working with Nonnegative Real Numbers

In mathematics, nonnegative real numbers include all positive numbers and zero. They are important when working with radicals because the principal square root, such as \(\sqrt{x}\), is defined only for nonnegative numbers in real number arithmetic.

• This ensures all base components in our calculation, like \(\sqrt{5}\) and \(\sqrt{12}\), are real and positive, guaranteeing meaningful results.

Nonnegative real numbers also imply no imaginary results that might complicate simplifying radical expressions. By using nonnegative real numbers, we ensure our math stays grounded in real world applications, aiding us in computing accurate and relatable solutions.