Problem 22 - Intermediate Algebra

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ -5 \sqrt{3}(3 \sqrt{12}-9 \sqrt{8}) $$

Step-by-Step Solution

Answer

The final expression is \(-90 + 90\sqrt{6}\).

Step 1: Distribute the Multiplication

Distribute the multiplication of \(-5\sqrt{3}\) with each term inside the parenthesis: \(-5\sqrt{3}\times 3\sqrt{12} - (-5\sqrt{3}\times 9\sqrt{8})\).This results in two expressions: \(-15\sqrt{3}\sqrt{12} + 45\sqrt{3}\sqrt{8}\).

Step 2: Simplify Each Square Root

Now, simplify the square roots: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\) and \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\). Replace these back into the expression: \(-15\sqrt{3}\times 2\sqrt{3} + 45\sqrt{3}\times 2\sqrt{2}\).

Step 3: Perform the Multiplications

Now, perform the multiplications: - For the first term, \(-15\sqrt{3} \times 2\sqrt{3} = -15 \times 2 \times \sqrt{3} \times \sqrt{3} = -30 \times 3 = -90\).- For the second term, \(45\sqrt{3} \times 2\sqrt{2} = 45 \times 2 \times \sqrt{3}\sqrt{2} = 90\sqrt{6}\).Combine these results: \(-90 + 90\sqrt{6}\).

Step 4: Write the Final Expression

Combine both simplified products from step 3 to form the final expression: \(-90 + 90\sqrt{6}\). This is the product expressed in the simplest radical form.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing or multiplying a term across terms inside parentheses. In the exercise we are tackling here, this concept plays a crucial role in addressing the expression \(-5 \sqrt{3}(3 \sqrt{12}-9 \sqrt{8})\).
Here's how it works:

• The term outside the parentheses, \(-5\sqrt{3}\),is applied to each term inside the parentheses, which are \(3\sqrt{12}\)and \(9\sqrt{8}\).
• This requires multiplying \(-5 \sqrt{3}\)each time separately by both \(3\sqrt{12}\)and \(-9\sqrt{8}\).
• This generates two new expressions: \(-15\sqrt{3}\sqrt{12} + 45\sqrt{3}\sqrt{8}\).

By breaking it down this way, the distributive property allows us to handle complex multiplication in a step-by-step manner, simplifying our calculations.

Simplest Radical Form

Working with radicals often requires expressing them in their simplest form, which is crucial for clear and concise solutions. In our example problem, the radicals \(\sqrt{12}\)and \(\sqrt{8}\)need simplification.
The goal is to find the largest perfect square factor in each radical:

• For \(\sqrt{12}\): Decompose as \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}\).
• For \(\sqrt{8}\): Decompose as \(\sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}\).

By simplifying like this, each radical is reduced to its simplest form, making them easier to work with. This method is often used in mathematics to make expressions more manageable and clearer, paving the way for the subsequent multiplication step.

Multiplying Radicals

Multiplying radicals can initially seem challenging, but it becomes manageable with proper strategy and understanding. In our specific case of simplification and multiplication, the process unfolds as follows:
First, we handle the simplified radicals obtained previously:

• For \(-15 \sqrt{3} \times 2 \sqrt{3}\): Multiply real numbers \(-15\times 2\) and radicals \(\sqrt{3} \times \sqrt{3}\) to produce \(-30 \times 3 = -90\).
• For \(45 \sqrt{3} \times 2 \sqrt{2}\): Multiply real numbers \(45 \times 2\) and radicals \(\sqrt{3} \times \sqrt{2}\) to yield \(90 \sqrt{6}\).

When multiplying radicals, always ensure:

• The coefficients (numbers outside the radicals) are multiplied separately from the radicals themselves.
• The product of like radicals results in the square of the number under the radical, simplifying further.

This approach, as illustrated in our example, ensures that expressions are always in their simplest radical form for effective and clear communication of results.