Problem 9 - Intermediate Algebra

Question

For Problems \(1-20\), use the distributive property to help simplify each of the following. For example, $$ \begin{aligned} 3 \sqrt{8}-\sqrt{32} &=3 \sqrt{4} \sqrt{2}-\sqrt{16} \sqrt{2} \\ &=3(2) \sqrt{2}-4 \sqrt{2} \\ &=6 \sqrt{2}-4 \sqrt{2} \\ &=(6-4) \sqrt{2}=2 \sqrt{2} \end{aligned} $$ $$ -9 \sqrt{24}+3 \sqrt{54}-12 \sqrt{6} $$

Step-by-Step Solution

Answer

The simplified expression is \(-21 \sqrt{6}\).

Step 1: Simplify Each Square Root

Start by simplifying the square roots where possible. Break down each square root into a product of perfect squares.\[-9 \sqrt{24} = -9 \sqrt{4 \times 6} = -9 \times 2 \sqrt{6} = -18 \sqrt{6}\] \[3 \sqrt{54} = 3 \sqrt{9 \times 6} = 3 \times 3 \sqrt{6} = 9 \sqrt{6}\] \[-12 \sqrt{6} \text{ is already in its simplest form.}\]

Step 2: Combine Like Terms

Now that all terms are expressed as some coefficient of \(\sqrt{6}\), combine the like terms.\[-18 \sqrt{6} + 9 \sqrt{6} - 12 \sqrt{6}\]Combine the coefficients: \(-18 + 9 - 12 = -21\). \[-21 \sqrt{6}\]

Key Concepts

Simplifying Square Roots

When tackling square roots, especially when they involve algebraic expressions, the first step is often simplifying them. Simplifying square roots involves expressing the square root as a product of its factors, specifically focusing on prime factors that are perfect squares.
Imagine you have \( \sqrt{24} \). You want to break this down into the product of numbers such that at least one is a perfect square. In this case, \(24\) can be factored into \(4 \times 6\), where \(4\) is a perfect square. This enables you to simplify as follows:

• \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}\)

By simplifying square roots, we make complex expressions easier to handle and facilitate further operations like addition or subtraction. This process lays the groundwork for combining any terms that may exist in your algebraic expression.

Like Terms

Like terms are terms in an algebraic expression that have the same variable components raised to the same power. In the context of simplifying expressions involving square roots, like terms are those terms that have the same radical part.
For the expression \(-9 \sqrt{24} + 3 \sqrt{54} - 12 \sqrt{6}\), after simplifying the square roots, you'll notice:

• \(-9 \sqrt{24} = -18 \sqrt{6}\)
• \(3 \sqrt{54} = 9 \sqrt{6}\)
• \(-12 \sqrt{6}\) is already simplified.

All terms are now like terms because they share the common factor of \(\sqrt{6}\). Recognizing like terms is crucial because it allows you to combine them by simply adding or subtracting their coefficients. You end up with \(-18 \sqrt{6} + 9 \sqrt{6} - 12 \sqrt{6}\), which simplifies to \(-21 \sqrt{6}\). Identifying and combining like terms is an essential skill in algebra to avoid mistakes and streamline calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can be simplified using various algebraic properties, such as the distributive property, to make them easier to understand or solve.
Let's consider the structure of the expression \(-9 \sqrt{24} + 3 \sqrt{54} - 12 \sqrt{6}\). Each term consists of a coefficient and a square root – an algebraic blend. The key to solving such expressions is to look for opportunities using the properties of numbers, like factoring and using perfect squares for roots.
Applying these principles, you simplify each term as much as possible, making sure to have common radical parts, helping in combining terms efficiently later. This process is all about turning a complex equation into something manageable, letting you focus on the arithmetic of the coefficients. Understanding and manipulating algebraic expressions is foundational in algebra and calculus, as it builds the connection between numerical calculations and abstract reasoning.