Problem 30 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(2 \sqrt{2}(3 \sqrt{12}-\sqrt{27})\)

Step-by-Step Solution

Answer

The product in simplest radical form is \(6 \sqrt{6}\).

Step 1: Simplify Each Radical

Begin by simplifying each term with radicals inside the parenthesis. For the term involving \(\sqrt{12}\), rewrite it as \(\sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\). Similarly, for \(\sqrt{27}\), rewrite it as \(\sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3}\). Thus, the expression simplifies to \(3 \times 2 \sqrt{3} - 3 \sqrt{3}\) which simplifies further to \(6 \sqrt{3} - 3 \sqrt{3}\).

Step 2: Perform the Operations

Now substitute the simplified radicals into the expression: \(2 \sqrt{2} (6\sqrt{3} - 3\sqrt{3})\) which simplifies to \(2 \sqrt{2} \times 3 \sqrt{3}\).

Step 3: Multiply the Constants and Radicals

Distribute \(2 \sqrt{2}\) across the simplified expression: calculate the multiplication of constants \(2 \times 3 = 6\) and the multiplication of the radicals \(\sqrt{2} \cdot \sqrt{3} = \sqrt{6}\). This gives \(6 \sqrt{6}\).

Step 4: Express in Simplest Form

The expression \(6 \sqrt{6}\) is already in its simplest form. Therefore, the product of the original expression is \(6 \sqrt{6}\).

Key Concepts

Multiplying Radicals

Multiplying radicals might look a bit intimidating at first, but it follows a simple set of rules. Just like regular multiplication, you multiply the numbers outside the radicals first, and then you multiply the numbers inside the radicals. Let's take the example from our problem:

• We have two components here: constants (the numbers outside the radicals) and the radicals themselves (the square root parts).
• We start by multiplying the constants outside the radicals. In our solution, this was the multiplication of 2 and 3, which gave us 6.
• Next, we multiply the numbers inside the radicals, i.e., \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \).

Combining these two steps gives us \( 6 \sqrt{6} \), which is your answer in a form that's often more straightforward to work with. Remember, it's important to have each component of your radicals simplified before multiplying, to ensure that you're getting the answer in its simplest form.

Simplest Radical Form

Simplifying radicals is akin to breaking down a complex problem into simpler parts, making it easier to solve. A radical is in its simplest form when there are no perfect square factors other than 1 in the radicand (the number inside the radical). Here's how you simplify radicals in a few steps:

• Identify any perfect square factors of the number inside the radical. For instance, in the case of \(\sqrt{12}\), we recognize \(4\) as a perfect square because \(4 = 2^2\).
• Rewrite \(\sqrt{12}\) as \( \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \).
• Do the same for other radicals as well, \(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\).

The goal is to make sure the number inside the square root has no perfect squares remaining.If there are no further simplifications possible, then the radical is in its simplest form, enabling easier multiplication or addition later.

Radical Expressions

Radical expressions involve numbers with square roots or other roots, and they can appear intimidating initially. They emerge often in geometry and science calculations, allowing us to represent roots neatly. Here’s what makes working with radicals straightforward once you get the hang of it:

• Work systematically by breaking down expressions into manageable parts.
• Always simplify your radicals before further calculation. This might involve factoring out perfect squares or other common factors.
• Don’t rush through these calculations; working at a steady pace helps in keeping errors at bay.

Understanding radical expressions can pave the way for mastering advanced math topics, as they serve as a crucial foundational concept. With practice, evaluating and simplifying these expressions becomes second nature, easing the way to explore more complex equations and higher-level math.