Problem 19 - Intermediate Algebra

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ 2 \sqrt{6}(3 \sqrt{8}-5 \sqrt{12}) $$

Step-by-Step Solution

Answer

24√3 - 60√2

Step 1: Distribute the terms

Let's distribute the term outside the parentheses, which is \(2 \sqrt{6}\), to each term inside the parentheses. First, distribute \(2 \sqrt{6}\) to \(3 \sqrt{8}\). Then distribute \(2 \sqrt{6}\) to \(-5 \sqrt{12}\):\[= 2 \sqrt{6} \times 3 \sqrt{8} - 2 \sqrt{6} \times 5 \sqrt{12}\]Now multiply the coefficients and radicals in each term separately.

Step 2: Multiply the coefficients and radicals

For the first multiplication, \(2 \sqrt{6} \times 3 \sqrt{8}\):- Multiply the coefficients: \(2 \times 3 = 6\).- Multiply the radicals: \(\sqrt{6} \times \sqrt{8} = \sqrt{48}\).This gives us the term \(6 \sqrt{48}\).For the second multiplication, \(2 \sqrt{6} \times 5 \sqrt{12}\):- Multiply the coefficients: \(2 \times 5 = 10\).- Multiply the radicals: \(\sqrt{6} \times \sqrt{12} = \sqrt{72}\).This gives us the term \(10 \sqrt{72}\).

Step 3: Simplify the square roots

Now simplify \(\sqrt{48}\) and \(\sqrt{72}\):- \(\sqrt{48}\): Factor 48 into 16 and 3: \(48 = 16 \times 3\), so \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \sqrt{3}\).- \(\sqrt{72}\): Factor 72 into 36 and 2: \(72 = 36 \times 2\), so \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6 \sqrt{2}\).

Step 4: Final simplification

Substitute the simplified square roots back into the expression:\[6 \sqrt{48} - 10 \sqrt{72} = 6 \times 4 \sqrt{3} - 10 \times 6 \sqrt{2}\]Which simplifies to:\[24 \sqrt{3} - 60 \sqrt{2}\].

Step 5: Solution Completion

The expression \(2 \sqrt{6}(3 \sqrt{8}-5 \sqrt{12})\) simplifies to \(24 \sqrt{3} - 60 \sqrt{2}\) in its simplest radical form.

Key Concepts

Radical Multiplication

When dealing with radical multiplication, the main idea is to multiply the numbers outside the square root symbols (the coefficients) with each other. Then, multiply the numbers inside the square root symbols (the radicands) as well. This process ensures that both parts of each radical expression are treated separately and correctly.

Here's a simple bullet list to guide through radical multiplication:

• First, multiply the coefficients outside the radicals.
• Next, multiply the radicands inside the radicals.
• Keep the result in the form of a coefficient multiplied by a radical.

For example, consider the multiplication of two radicals: \[3 \sqrt{5} \times 2 \sqrt{10}\].
Here, multiply the coefficients: \(3 \times 2 = 6\). Then multiply the radicands: \(\sqrt{5} \times \sqrt{10} = \sqrt{50}\). The result is \(6 \sqrt{50}\).

This step is important because it lays the groundwork for further simplification, allowing the expression to be rewritten in its simplest form.

Square Roots Simplification

Simplifying square roots involves reducing a square root to its simplest form. The key is to break down the number inside the square root (the radicand) into its prime factors, and pair them to pull outside the radical.

Let's illustrate this with an example:
Take \(\sqrt{48}\). Break down 48 into its prime factorization:
\(48 = 2 \times 2 \times 2 \times 2 \times 3\).
Pair the twos to move a 2 out of the radical for each pair: \(\sqrt{48} = \sqrt{(2 \times 2) \times (2 \times 2) \times 3} = 4 \sqrt{3}\).

Important steps in square root simplification include:

• Identifying pairs of factors inside the radical.
• Pulling one number out from the radical for each pair.
• Leaving any unpaired numbers under the square root.

This method transforms complex numbers under the square root sign into simpler terms, making it easier to work with radical expressions in calculations and simplifications.

Distributive Property

The distributive property is a foundational algebraic principle that allows you to multiply a single term across terms inside parentheses. This property is essential when dealing with expressions that involve multiplication of terms with radicals.

Consider the expression: \(a (b + c)\).
Using the distributive property, this becomes:
\(a \times b + a \times c\).
The same rule applies when working with radicals. For example, consider the expression \(2 \sqrt{6}(3 \sqrt{8} - 5 \sqrt{12})\).

Apply the distributive property:

• Multiply \(2 \sqrt{6}\) by \(3 \sqrt{8}\).
• Then multiply \(2 \sqrt{6}\) by \(-5 \sqrt{12}\).

By distributing the radical term in front across the expression inside the parentheses, the distributive property helps in breaking down and simplifying expressions involving radicals in an efficient and structured way.