Problem 21 - Algebra and Trigonometry

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 for Radicals

You can rewrite the given expression \(\sqrt{2 x^{2}} \cdot \sqrt{6 x}\) by applying the product rule for radicals to multiply the numbers and variables under the square root symbol. This will give: \(\sqrt{(2 x^{2}) \cdot (6 x)}\)

Step 2: Simplify the Expression inside the Square Root

You can simplify the product inside the square root: \(2 x^{2} \cdot 6 x\) which gives \(12x^{3}\). So the expression will become: \(\sqrt{12x^{3}}\)

Step 3: Factor and Simplify the Square Root

You can rewrite \(12x^{3}\) as \(4\cdot3\cdot(x^{2}\cdot x)\). Since the square root of \(4\) and \(x^{2}\) are integers, you can simplify the expression to \(2x\sqrt{3x}\)

Key Concepts

Simplifying Radicals

Understanding how to simplify radicals is important when dealing with expressions that involve roots, particularly square roots. The goal is to make the expression as simple as possible, by combining like terms and factoring. This process often involves identifying numbers and variables that can be removed from under the root by exploiting their perfect powers.

The simplification process works by:

• Identifying perfect squares or cubes within the expression.
• Removing these from under the radical since the square root of a perfect square is a whole number.
• Once extracted, these can be simplified further as a regular multiplication outside the root.

This step-by-step simplification is crucial for making complex mathematics easier and more intuitive, ensuring accuracy in answers when solving problems with radicals.

Square Root

Square roots are a way to express the value that, when multiplied by itself, gives the original number. The symbol \( \sqrt{} \) is used to denote a square root. Taking the square root of a number is essentially the opposite of squaring that number.

Here’s how square roots function:

• If you have \( \sqrt{9} \), the answer is \(3\) because \(3 \times 3 = 9\).
• Square roots can be applied to any positive number or zero, and for the purpose of this exercise, we consider nonnegative real numbers.
• The process of simplifying square roots involves estimating or calculating the exact square root, especially for perfect squares.

Finding square roots is fundamental when working with radical expressions because it enables you to take a complex number under the root and find a simpler whole number whenever possible.

Multiplication of Radicals

The multiplication of radicals, and specifically square roots, can be tackled using the Product Rule for Radicals. This rule allows you to multiply two radicals as long as their indices are the same.

For instance:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)
• This operation combines the numbers under a single radical symbol, which can be simplified further if possible.
• The rule applies universally to numbers and variables, helping simplify expressions quickly and efficiently.

When executed correctly, multiplying radicals shrinks the expression size and makes it more manageable, often allowing easier progression through algebraic equations or other complex problems.