Solving $(-3 \sqrt{5})$ $(6 \sqrt{2})$ A Step-by-Step Guide

JU07/09/2025 08, 2025 by THE IDEN 60 views

$Unveiling the Solution to $(-3 \sqrt{5})$ $(6 \sqrt{2})$: A Comprehensive Guide$

Navigating the realm of mathematical expressions, particularly those involving radicals, can sometimes feel like traversing a complex maze. However, with a systematic approach and a clear understanding of the fundamental principles, even the most intricate problems can be demystified. In this comprehensive guide, we will embark on a journey to unravel the solution to the expression $(-3 \sqrt{5})$ $(6 \sqrt{2})$ , providing a step-by-step analysis and illuminating the underlying concepts along the way.

Deciphering the Expression: A Step-by-Step Breakdown

At first glance, the expression $(-3 \sqrt{5})$ $(6 \sqrt{2})$ may appear daunting, but by breaking it down into smaller, manageable components, we can gain a clearer perspective. This expression involves the multiplication of two terms, each containing a combination of integers and radicals. Our primary objective is to simplify this expression by performing the necessary mathematical operations.

1. Identifying the Components

The first step in our analytical process involves identifying the individual components of the expression. We can observe that the expression consists of two terms: $(-3 \sqrt{5})$ and $(6 \sqrt{2})$ . Each of these terms is composed of an integer coefficient and a radical term. In the first term, the integer coefficient is -3, and the radical term is $\sqrt{5}$ . Similarly, in the second term, the integer coefficient is 6, and the radical term is $\sqrt{2}$ .

2. Applying the Multiplication Rule

The core principle that governs the simplification of this expression is the multiplication rule for radicals. This rule states that the product of two radicals is equal to the radical of the product of the radicands. In mathematical notation, this can be expressed as: $\sqrt{a} * \sqrt{b} = \sqrt{a * b}$ , where 'a' and 'b' are non-negative real numbers. This rule serves as the cornerstone of our simplification process.

3. Multiplying the Integer Coefficients

Before we delve into the multiplication of the radical terms, let's first address the integer coefficients. We have -3 and 6 as our integer coefficients. Multiplying these two numbers together, we get: -3 * 6 = -18. This result will form the integer component of our final simplified expression.

4. Multiplying the Radical Terms

Now, let's turn our attention to the radical terms: $\sqrt{5}$ and $\sqrt{2}$ . Applying the multiplication rule for radicals, we multiply the radicands (the numbers inside the square roots) together: 5 * 2 = 10. Therefore, the product of the radical terms is $\sqrt{10}$ .

5. Combining the Results

Having computed the product of the integer coefficients and the product of the radical terms, we can now combine these results to arrive at the final simplified expression. We have -18 as the product of the integer coefficients and $\sqrt{10}$ as the product of the radical terms. Multiplying these together, we get: -18 * $\sqrt{10}$ = -18 $\sqrt{10}$ .

The Solution: $-18 \sqrt{10}$

Therefore, after a meticulous step-by-step analysis, we have arrived at the solution to the expression $(-3 \sqrt{5})$ $(6 \sqrt{2})$ . The simplified form of this expression is $-18 \sqrt{10}$ , which corresponds to option A in the given choices. This solution encapsulates the application of fundamental mathematical principles, including the multiplication rule for radicals and the arithmetic operations involving integers.

Deeper Dive: Understanding Radicals and Their Properties

To truly grasp the essence of simplifying expressions involving radicals, it's crucial to delve deeper into the concept of radicals themselves and their inherent properties. Radicals, often denoted by the square root symbol ( $\sqrt{}$ ), represent the inverse operation of exponentiation. In simpler terms, the square root of a number 'x' is a value that, when multiplied by itself, equals 'x'.

Types of Radicals

While square roots are the most commonly encountered type of radicals, the concept extends to higher-order roots as well. Cube roots, denoted by $\sqrt[3]{}$ , represent the value that, when multiplied by itself three times, equals the radicand. Similarly, fourth roots, fifth roots, and so on, can be defined. The general form of a radical is $\sqrt[n]{x}$ , where 'n' is the index of the radical (indicating the order of the root) and 'x' is the radicand.

Simplifying Radicals

Simplifying radicals involves expressing them in their simplest possible form. This typically entails identifying perfect square factors within the radicand and extracting their square roots. For instance, $\sqrt{12}$ can be simplified as follows:

Identify the perfect square factors of 12: 12 = 4 * 3, where 4 is a perfect square (2 * 2).
Rewrite the radical: $\sqrt{12}$ = $\sqrt{4 * 3}$ .
Apply the multiplication rule for radicals: $\sqrt{4 * 3}$ = $\sqrt{4}$ * $\sqrt{3}$ .
Simplify: $\sqrt{4}$ * $\sqrt{3}$ = 2 $\sqrt{3}$ .

Therefore, the simplified form of $\sqrt{12}$ is 2 $\sqrt{3}$ .

Operations with Radicals

Radicals can be subjected to various mathematical operations, including addition, subtraction, multiplication, and division. However, certain rules and considerations must be adhered to when performing these operations.

Addition and Subtraction

Radicals can only be added or subtracted if they have the same radicand. For instance, 2 $\sqrt{5}$ + 3 $\sqrt{5}$ can be simplified to 5 $\sqrt{5}$ because both terms have the same radicand, $\sqrt{5}$ . However, 2 $\sqrt{5}$ + 3 $\sqrt{2}$ cannot be simplified further because the radicands are different.

Multiplication

The multiplication of radicals, as we have seen in the solution to our initial expression, follows the multiplication rule: $\sqrt{a}$ * $\sqrt{b}$ = $\sqrt{a * b}$ . This rule allows us to combine radicals under a single radical sign.

Division

The division of radicals can be performed by dividing the radicands, provided the radicals have the same index. For instance, $\frac{\sqrt{12}}{\sqrt{3}}$ = $\sqrt{\frac{12}{3}}$ = $\sqrt{4}$ = 2.

Conclusion: Mastering Radicals for Mathematical Proficiency

In conclusion, the solution to the expression $(-3 \sqrt{5})$ $(6 \sqrt{2})$ is $-18 \sqrt{10}$ , which corresponds to option A. This solution not only provides the correct answer but also serves as a gateway to a deeper understanding of radicals and their properties. By grasping the fundamental concepts of radicals, their simplification techniques, and the rules governing their operations, you can enhance your mathematical proficiency and confidently tackle a wide range of problems involving radicals. The journey through the realm of radicals may present its challenges, but with a systematic approach and a thirst for knowledge, you can unlock the beauty and power of these mathematical entities.

This exploration into the world of radicals should empower you to approach similar problems with greater confidence and understanding. Remember, mathematics is a journey of discovery, and every solved problem is a step forward on that path.