Problem 21 - College Algebra

Question

Use the product rule to simplify the expressions in Exercises \(13-22\). In Exercises \(17-22,\) assume that variables represent nonnegative real numbers. $$\sqrt{2 x^{2}} \cdot \sqrt{6 x}$$

Step-by-Step Solution

Answer

The simplified expression is \(2x^{2}\sqrt{3}\).

Step 1: Break Down the Expression

First, you need to decompose each square root into its simplest form. This implies that \(\sqrt{2 x^{2}}\) will become \(x\sqrt{2}\) and \(\sqrt{6x}\) turns into \(\sqrt{6}\sqrt{x}\).

Step 2: Apply the Product Rule

The product rule for roots specifies that the product of two radical expressions of the same radical type can be written as a single radical expression. This implies \(\sqrt{a}\sqrt{b}\) can be written as \(\sqrt{ab}\). Apply this rule to the decomposed expression by multiplying \(x\sqrt{2}\) and \(\sqrt{6}\sqrt{x}\), resulting in \(x\sqrt{2}\cdot \sqrt{6}\cdot x\). It can then be simplified to \(x^{2}\sqrt{12}\).

Step 3: Simplify the Result

Further simplify \(x^{2}\sqrt{12}\) where you simplify the constant under the radical to \(\sqrt{12}\) = \(2\sqrt{3}\). Finally, the expression simplifies to \(2x^{2}\sqrt{3}\).

Key Concepts

Understanding Radical Expressions

Radical expressions are mathematical expressions that involve roots, most commonly square roots. A radical expression is written with the radical sign \( \sqrt{} \), and the number or expression under the radical sign is called the radicand.
For example, in the expression \( \sqrt{2x^2} \), \(2x^2\) is the radicand.When working with radical expressions, particularly when simplifying them, the goal is to express them in their simplest form. This can involve:

• Reducing the radicand to its most basic components.
• Combining radical expressions, when possible, using rules like the product rule.
• Ensuring that there are no radicals in the denominator (rationalizing the denominator if necessary).

To simplify most radical expressions properly, it is important to understand the properties of roots and the rules for working with them. This foundational understanding will make working with any complex radical expression much more manageable.

Mastering Square Root Simplification

Square root simplification revolves around expressing a given square root in its simplest form. To accomplish this, you need to identify and reduce any perfect squares within the radicand.
Suppose you have \(\sqrt{12}\). Begin by factoring \(12\) into its prime factors: \(12 = 2^2 \times 3\). The perfect square here is \(2^2\), which can be extracted, simplifying the expression to \(2\sqrt{3}\).The process involves:

• Identifying any perfect squares in the radicand.
• Extracting these perfect squares out of the radical.
• Simplifying the expression to its simplest terms.

Remember: The aim is to express the square root or any radical expression in a way that reveals its simplified nature without altering the expression's value. This makes comparing and calculating with these expressions much more straightforward.

Effective Algebraic Simplification Techniques

Algebraic simplification involves reducing mathematical expressions to their most efficient forms, making them easier to work with. It could involve basic operations, factoring, or using specific rules such as distributing or combining like terms.When simplifying expressions like \(x^2\sqrt{12}\) to \(2x^2\sqrt{3}\), things to keep in mind include:

• Ensuring all like terms are combined and all possible factors are factored out.
• Using rules such as the distributive property to combine terms effectively.
• Recognizing equivalent expressions that appear in different forms and simplifying them accordingly.

By practicing algebraic simplification, students not only solve problems more efficiently but also develop a deeper understanding of how mathematical operations interplay with one another. This skill is fundamental in advancing mathematical knowledge and facilitating successful problem-solving across varied topics.