Problem 21 - College Algebra Essentials

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 form of the given expression \( \sqrt{2 x^{2}} \cdot \sqrt{6 x} \) is \(2x \sqrt{3x}\)

Step 1: Apply the product rule

Combine the two square root expressions under a single square root sign using the product rule: \( \sqrt {2x^2 \cdot 6x} \)

Step 2: Consolidate the expression inside the square root

Multiply the numbers and the variables separately: \( \sqrt{12x^3} \)

Step 3: Further simplify the square root term

The square root of 12 can be simplified to \(2 \sqrt{3}\) and the square root of \(x^3\) can be simplified to \(x \sqrt{x}\) so the final simplified form of the expression is: \( 2x \sqrt{3x} \)

Key Concepts

Simplify Radical Expressions

Radical expressions, or expressions that involve square roots, can often seem intimidating, but their simplification is guided by a set of rules that, once understood, can make the process much more approachable. Simplifying radical expressions involves identifying and extracting perfect square factors, and combining like terms under a single radical whenever possible.

For example, consider the radical expression \( \sqrt{12x^3} \). To simplify, you would first look at the number 12 and the variable term \(x^3\) separately. Since 12 equals \(2^2 \times 3\), and the exponent in \(x^3\) indicates that there are three \(x\)'s multiplied together, we can extract one \(x\) as the square root of \(x^2\), thus simplifying to \(2x \sqrt{3x}\).

By systematically breaking down the expression into its constituent parts, we can turn a complicated problem into a series of simpler steps.

Applying Product Rule in Algebra

The product rule is a fundamental concept in algebra that comes in handy when multiplying radical expressions. Essentially, the product rule allows you to combine two square roots into one by multiplying their contents. The rule states that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\), where \(a\) and \(b\) are any nonnegative real numbers.

In practice, this rule means that you can combine the contents of square roots before simplifying, which often leads to a more streamlined and efficient solution. As seen in the exercise \(\sqrt{2x^2} \cdot \sqrt{6x}\), applying the product rule first gives us a single radical, \(\sqrt{12x^3}\), making it easier to simplify further.

Square Root Multiplication

Multiplying square roots might seem perplexing at first, but it can be made clear by using the product rule of square roots. When you multiply square roots, you're essentially multiplying the numbers inside the roots and then taking the square root of the resulting product.

However, it's important to remember that this only works straightforwardly for nonnegative numbers, as the rules for square roots of negative numbers (which involve imaginary numbers) are different. Keeping this in mind, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) simplifies the process of multiplying square roots and is essential to transforming complex radical expressions into more manageable forms.

Radical Expression Simplification

To simplify radical expressions, identifying perfect squares within the radicand—the number under the square root—is essential. A perfect square is a number that is the square of an integer. When such a number is found, it can be taken out of the square root, making the expression simpler.

Additionally, variables within the radicand that are raised to an even power can also be simplified; for instance, \(x^2\) under a square root becomes just \(x\). By looking for these opportunities to simplify within radical expressions, students can break down complex problems into easier-to-manage pieces, leading to a clearer understanding and neater solutions.