Problem 22 - Algebra for College Students

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 simplest radical form is \\(-90 + 90\sqrt{6}\\).

Step 1: Distribute the outside term

First, distribute the term \(-5 \sqrt{3}\) across the expression inside the parenthesis, which is \(3 \sqrt{12} - 9 \sqrt{8}\). This gives us two separate products: \(-5 \sqrt{3} \times 3 \sqrt{12}\) and \(-5 \sqrt{3} \times -9 \sqrt{8}\).

Step 2: Simplify each product individually

For the first product, calculate \((-5) \times 3 = -15\) and multiply the square roots: \(\sqrt{3} \cdot \sqrt{12} = \sqrt{36}\). Hence, the first product becomes \(-15 \cdot \sqrt{36}\). The second product involves \((-5) \times (-9) = 45\) and \(\sqrt{3} \cdot \sqrt{8} = \sqrt{24}\). So the second product becomes \(45 \cdot \sqrt{24}\).

Step 3: Simplify the square roots

Compute \(\sqrt{36} = 6\), turning the first expression into \(-15 \times 6 = -90\). For \(\sqrt{24}\), simplify it as follows: \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}\). So the second expression is \(45 \times 2 \sqrt{6} = 90\sqrt{6}\).

Step 4: Combine simplified expressions

Combine the results from Steps 2 and 3: \(-90\) and \(90 \sqrt{6}\), resulting in the simplified expression: \(-90 + 90\sqrt{6}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, often used to simplify expressions and solve equations. When dealing with an expression like \(-5 \sqrt{3}(3 \sqrt{12}-9 \sqrt{8})\), the distributive property allows you to multiply the term outside the parenthesis by each term inside.

Think of it like distributing items into boxes: each item in the outer box needs to go into every inner box.

• In this exercise, \(-5 \sqrt{3}\) is multiplied by both \(3 \sqrt{12}\) and \(-9 \sqrt{8}\),resulting in two separate products.
• Don't forget to consider the signs: a negative term distributed into a negative term makes it positive.

Using this property helps to break down and simplify complex radical expressions step by step, making them easier to work with.

Radical Multiplication

Radical multiplication is the process of multiplying numbers under the radical sign. To multiply radicals, you multiply the numbers outside the radical and the numbers inside separately.

This means that for \(-5 \sqrt{3} \times 3 \sqrt{12}\), you first multiply \(-5\) and \(3\) to get \(-15\), and then multiply the radicals: \(\sqrt{3} \cdot \sqrt{12} = \sqrt{36}\).

• For radicals, the product inside is simplified to a single square root.
• Don't forget, if \(a \cdot a = a^2\), then \(\sqrt{a} \cdot \sqrt{a} = a\).

Remember, simplifying each part separately helps in keeping track of your progress and avoiding mistakes.

Simplifying Square Roots

Simplifying square roots involves finding the simplest form of a radical expression.

In the example, \(\sqrt{36}\) simplifies directly to \(6\) because \(36\) is a perfect square.

• To simplify \(\sqrt{24}\), break it into factors: \(\sqrt{4 \times 6}\).
• The square root of \(4\) is \(2\), so it becomes \(2\sqrt{6}\).

Finding perfect square factors aids in breaking down more complicated expressions. As a tip, always look to factor the number under the radical into two parts—one of which should be a perfect square. This can greatly simplify the final expression.