Simplifying Radicals Step-by-Step Guide To Solving $\frac{(10 \sqrt{2})(-4 \sqrt{6})}{60 \sqrt{3}}$

Introduction to Simplifying Radicals

When dealing with mathematical expressions involving radicals, it's often necessary to simplify them to their simplest form. Simplifying radicals not only makes the expressions easier to understand but also facilitates further calculations. In this comprehensive guide, we will delve into the process of simplifying the expression $\frac{(10 \sqrt{2})(-4 \sqrt{6})}{60 \sqrt{3}}$, breaking it down step by step to ensure clarity and understanding. Our main keyword here is simplifying radicals, and we will explore various techniques and properties that come into play when handling such expressions.

The journey of simplifying radicals begins with understanding the fundamental properties of square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. However, expressions involving square roots can often be simplified if the number under the radical (the radicand) has factors that are perfect squares. A perfect square is a number that can be obtained by squaring an integer, such as 1, 4, 9, 16, and so on. By identifying these factors, we can extract their square roots and simplify the expression.

The process of simplifying radicals is not just a mathematical exercise; it is a critical skill in various fields of science and engineering. From calculating the length of the diagonal of a square to determining the energy levels of electrons in an atom, radicals appear in many real-world applications. Therefore, mastering the art of simplifying radicals is an invaluable asset for anyone pursuing a career in these areas. This guide aims to provide a thorough understanding of the techniques involved, ensuring that you can confidently tackle any radical simplification problem.

Step-by-Step Simplification of $\frac{(10 \sqrt{2})(-4 \sqrt{6})}{60 \sqrt{3}}$

To simplify the expression $\frac{(10 \sqrt{2})(-4 \sqrt{6})}{60 \sqrt{3}}$, we will proceed systematically, breaking it down into manageable steps. This will not only help in understanding the process but also ensure accuracy in the final result. Our approach will involve several key techniques, including multiplying radicals, simplifying square roots, and rationalizing the denominator. The main focus here is on simplifying radicals, and we will demonstrate how each step contributes to achieving the simplest form.

1. Multiplying the Radicals in the Numerator

The first step in simplifying radicals is to multiply the terms in the numerator. We have $(10 \sqrt{2})(-4 \sqrt{6})$. To multiply these terms, we multiply the coefficients (the numbers outside the square roots) and the radicands (the numbers inside the square roots) separately. This gives us:

$$(10 \times -4) \times (\sqrt{2} \times \sqrt{6}) = -40 \sqrt{2 \times 6} = -40 \sqrt{12}$$

So, the numerator becomes $-40 \sqrt{12}$. This step is crucial because it combines the radicals into a single term, making it easier to identify potential simplifications. Understanding how to multiply radicals is fundamental to simplifying radicals, and this step illustrates the application of the property $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.

2. Simplifying the Square Root in the Numerator

Now we need to simplify $\sqrt{12}$. To do this, we look for perfect square factors of 12. The factors of 12 are 1, 2, 3, 4, 6, and 12. Among these, 4 is a perfect square (since $2^2 = 4$). We can rewrite 12 as $4 \times 3$. Thus, we have:

$$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$

Substituting this back into our numerator, we get:

$$-40 \sqrt{12} = -40 \times 2\sqrt{3} = -80\sqrt{3}$$

This step highlights the importance of identifying perfect square factors when simplifying radicals. By extracting the square root of the perfect square, we reduce the radicand to its smallest possible value, which is a key aspect of simplification.

3. Rewriting the Expression

Now that we've simplified the numerator, we can rewrite the entire expression as:

$$\frac{-80\sqrt{3}}{60\sqrt{3}}$$

This form allows us to see the expression more clearly and identify further simplifications. The main keyword simplifying radicals is evident here, as we've transformed the original complex expression into a simpler fraction involving square roots.

4. Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 80 and 60 is 20. Dividing both numbers by 20, we get:

$$\frac{-80}{60} = \frac{-80 \div 20}{60 \div 20} = \frac{-4}{3}$$

Now, our expression becomes:

$$\frac{-4\sqrt{3}}{3\sqrt{3}}$$

This step demonstrates how simplifying the numerical part of the expression contributes to the overall simplification. The keyword simplifying radicals is still central here, as we're reducing the fraction to its lowest terms.

5. Cancelling the Common Radical

We can see that $\sqrt{3}$ appears in both the numerator and the denominator. Since $\frac{\sqrt{3}}{\sqrt{3}} = 1$, we can cancel out the common radical:

$$\frac{-4\sqrt{3}}{3\sqrt{3}} = \frac{-4}{3} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{-4}{3} \times 1 = -\frac{4}{3}$$

Thus, the simplified expression is $-\frac{4}{3}$. This final step showcases the elegance of simplifying radicals by eliminating common factors. By removing the radical from the expression, we have arrived at the simplest form.

Conclusion on Simplifying Radicals

In conclusion, simplifying radicals involves a series of steps that include multiplying radicals, identifying and extracting perfect square factors, simplifying fractions, and canceling common radicals. By following these steps systematically, we can reduce complex radical expressions to their simplest forms. In the case of the expression $\frac{(10 \sqrt{2})(-4 \sqrt{6})}{60 \sqrt{3}}$, we have shown how each step contributes to the final simplified result of $-\frac{4}{3}$. This process is not only mathematically sound but also provides a clear and understandable way to handle radical expressions.

The ability to simplify radicals is a valuable skill in mathematics and its applications. It allows for more concise and manageable expressions, which are essential for further calculations and problem-solving. This guide has provided a detailed walkthrough of the simplification process, emphasizing the key techniques and properties involved. By mastering these techniques, you can confidently approach any radical simplification problem and achieve the simplest form.

Understanding and applying these principles of simplifying radicals not only enhances mathematical proficiency but also provides a foundation for advanced concepts in algebra and calculus. The steps outlined in this guide serve as a comprehensive resource for anyone looking to improve their skills in this area. Through consistent practice and application of these techniques, the process of simplifying radicals becomes second nature, making complex mathematical problems more accessible and solvable.