Problem 20 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(4 \sqrt{2}(3 \sqrt{12}+7 \sqrt{6})\)

Step-by-Step Solution

Answer

The product in simplest radical form is \(24\sqrt{6} + 56\sqrt{3}\).

Step 1: Distribute the Outer Term

Begin by distributing the term outside the parentheses, which is \(4\sqrt{2}\), to each term inside the parentheses: \(3\sqrt{12}\) and \(7\sqrt{6}\). This gives us: \[4\sqrt{2} \cdot 3\sqrt{12} + 4\sqrt{2} \cdot 7\sqrt{6}.\] The expression can be rewritten as: \[12\sqrt{24} + 28\sqrt{12}.\]

Step 2: Simplify the Radicals

Simplify \(\sqrt{24}\) and \(\sqrt{12}\) into their simplest forms using their prime factorizations: \(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\) and \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\). Substitute these simplified forms back into the expression: \[12(2\sqrt{6}) + 28(2\sqrt{3}),\] which becomes \[24\sqrt{6} + 56\sqrt{3}.\]

Step 3: Final Simplified Expression

The expression \(24\sqrt{6} + 56\sqrt{3}\) is already in its simplest radical form as there are no further common factors or simplifications possible when adding the terms. The final answer is therefore \(24\sqrt{6} + 56\sqrt{3}\).

Key Concepts

The Distributive Property

Let's talk about the distributive property, a handy tool in algebra. Imagine you have to handle terms both inside and outside the parentheses. The distributive property helps by telling you to distribute, or "pass out," the term outside to every term inside the parentheses.

In our example, we have an expression: \(4 \sqrt{2}(3 \sqrt{12}+7 \sqrt{6})\). The \(4\sqrt{2}\) is on the outside, and we distribute it to both \(3\sqrt{12}\) and \(7\sqrt{6}\).

First, multiply \(4\sqrt{2}\) by \(3\sqrt{12}\), which gives \(12\sqrt{24}\).

Next, multiply \(4\sqrt{2}\) by \(7\sqrt{6}\), giving \(28\sqrt{12}\).

By distributing these terms, you have now laid the groundwork to simplify further. This powerful property lets you spread out your calculations to make things simpler and more manageable.

Simplifying Radicals

After using the distributive property, the next step is simplifying the radicals. Think of this as turning complex roots into something more recognizable. Radicals are based on the root concept in mathematics, like square roots.

Take \(\sqrt{24}\) and \(\sqrt{12}\) in our expression. These can be broken down using prime numbers.

\(\sqrt{24}\) is equivalent to \(\sqrt{4 \times 6}\), simplifying further to \(2\sqrt{6}\).

\(\sqrt{12}\) translates to \(\sqrt{4 \times 3}\), reducing to \(2\sqrt{3}\).

Substitute these back into the expression to achieve a simpler form: \(24\sqrt{6} + 56\sqrt{3}\). Now, the radicals are cleaner, and the expression is more straightforward. This step is crucial because it uncovers and clarifies the underlying simpler radicals hiding within.

Prime Factorization

Prime factorization is like uncovering the basic building blocks of a number. When simplifying radicals, prime factorization shows you exactly how to break down numbers to their core components.

Every number has prime factors. Consider \(24\) and \(12\) from our example:

\(24\) can be expressed as \(2 \times 2 \times 2 \times 3\).

\(12\) becomes \(2 \times 2 \times 3\).

Using these prime factors helps simplify radicals effectively.
When you know that \(\sqrt{24}\) breaks into \(\sqrt{4 \times 6}\) using prime factors, you can easily spot that \(\sqrt{4}\) equals \(2\). This same approach applies to \(\sqrt{12}\), reducing it through \(\sqrt{4}\), to a simpler \(2\sqrt{3}\).
Prime factorization is all about making the unseen, seen. It reveals the smallest elements of numbers, allowing you to simplify complex expressions confidently.