Problem 15 - College Algebra

Question

Use the product rule to simplify the expressions in Exercises \(7-16 .\) In Exercises \(11-16,\) assume that variables represent nonnegative real numbers. $$\sqrt{2 x^{2}} \cdot \sqrt{6 x}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{2 x^{2}} \cdot \sqrt{6 x}\) is \(2x*\sqrt{3x}\)

Step 1 Apply the Product Rule

The product rule for radicals states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). So applying this rule to the given expression, we have \(\sqrt{2 x^{2}} \cdot \sqrt{6 x} = \sqrt{(2 x^{2}) (6 x)}\)

Step 2 Expand and Simplify

Next, multiply the contents of the root, which gives: \(\sqrt{(2 x^{2}) (6 x)} = \sqrt{12 x^{3}}\)

Step 3 Simplify the Root

The expression inside the root, \(12 x^{3}\) can be simplified as \(\sqrt{4*3*(x^{2} * x)}\), which simplifies to \(\sqrt{(4x^2)}*\sqrt{3x} = 2x*\sqrt{3x}\).

Key Concepts

Understanding Radical Expressions

Radical expressions are essentially expressions that include a symbol called the radical sign (√). This sign represents the root of a number. One of the most commonly encountered radicals is the square root. For example, in the expression \( \sqrt{2x^2} \cdot \sqrt{6x} \), the square root sign is used to denote the square roots of the expressions inside.

Radical expressions can appear complex, but by understanding their properties, we can easily manipulate and simplify them.

• The expression inside the radical sign is called the radicand.
• The radical sign itself indicates the root. Without a specified number, it's typically the square root.
• In algebra, radicals allow us to express roots of quantities that may not always resolve into whole numbers.

As you practice, you'll learn how to apply rules like the product rule for radicals to simplify these kinds of expressions efficiently.

Steps to Simplifying Expressions

Simplifying expressions means rewriting them in the simplest form without changing their value. When it comes to radical expressions, this usually involves reducing the radicand when possible and eliminating radicals from the denominator.

Here are some basic steps to follow:

• Identify and apply relevant rules, such as the product rule for radicals, to combine and simplify radical expressions.
• Look for factors within the radicand that are perfect squares (or cubes, if it’s a cube root) because they can be taken out of the radical.
• Simplify the expression by multiplying or dividing both inside and outside of the radicals as applicable.

Let's take a look at our example: \( \sqrt{2x^2} \cdot \sqrt{6x} = \sqrt{12x^3} \). By identifying that \(12x^3\) includes a perfect square factor, \(4x^2\), we can take \(2x\) out of the radical, thereby simplifying the expression to \(2x\sqrt{3x}\). This process not only makes expressions appear neater but often aids in further calculations.

Basic Algebraic Simplification Techniques

Algebraic simplification involves reducing expressions to their simplest form using various algebraic rules and properties. In the context of radical expressions, simplification looks much like what we’ve demonstrated—but let’s delve deeper into the underlying techniques.

Key algebraic techniques that help in simplification include:

• Factoring: Breaking down expressions into products of simpler factors, such as finding perfect square factors within radicands.
• Combining Like Terms: Terms that are alike are added or subtracted to streamline expressions.
• Using the Distributive Property: Applying this property to multipliers helps in opening or factoring expressions as needed.

By practicing these techniques, you'll become more adept at unraveling complex expressions into more workable forms. For example, in the simplification from \( \sqrt{12x^3} \) to \(2x\sqrt{3x}\), identifying the perfect squares and simplifying them was key to achieving the final result. Regularly engaging with these strategies will sharpen your algebraic prowess, paving the way for easier manipulation of more advanced expressions in the future.