Simplifying The Expression (-9√2)(4√6) A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to represent complex mathematical statements in a more concise and understandable form. This article delves into the process of simplifying the expression (-9√2)(4√6), providing a step-by-step guide that will not only lead you to the correct answer but also enhance your understanding of radical operations.

Understanding the Basics of Radical Simplification

Before we embark on the journey of simplifying the given expression, it's crucial to grasp the core concepts of radical simplification. A radical, denoted by the symbol √, represents the root of a number. For instance, √9 represents the square root of 9, which is 3, because 3 multiplied by itself equals 9. Similarly, √8 represents the square root of 8, which is approximately 2.828.

Simplifying radicals involves expressing them in their simplest form, where the radicand (the number under the radical sign) has no perfect square factors other than 1. For example, √8 can be simplified as √(4 * 2) = √4 * √2 = 2√2, since 4 is a perfect square (2 * 2 = 4). This simplification process makes it easier to perform operations involving radicals, such as multiplication and division.

Key Concepts for Radical Simplification:

  • Product Property of Radicals: √(a * b) = √a * √b
  • Quotient Property of Radicals: √(a / b) = √a / √b
  • Simplifying Radicals with Perfect Square Factors: √a² = a

Step-by-Step Simplification of (-9√2)(4√6)

Now that we have a solid foundation in radical simplification, let's tackle the expression (-9√2)(4√6) step by step:

Step 1: Apply the Commutative and Associative Properties of Multiplication

The commutative property states that the order in which we multiply numbers does not affect the result (a * b = b * a). The associative property states that the way we group numbers in multiplication does not affect the result (a * (b * c) = (a * b) * c). Applying these properties, we can rearrange and regroup the terms in the expression:

(-9√2)(4√6) = (-9 * 4)(√2 * √6)

Step 2: Multiply the Coefficients

The coefficients are the numbers in front of the radicals. In this case, the coefficients are -9 and 4. Multiplying them together, we get:

-9 * 4 = -36

So, the expression now becomes:

-36(√2 * √6)

Step 3: Apply the Product Property of Radicals

The product property of radicals states that the square root of a product is equal to the product of the square roots (√(a * b) = √a * √b). Applying this property, we can combine the radicals:

√2 * √6 = √(2 * 6) = √12

Now, the expression is:

-36√12

Step 4: Simplify the Radical √12

To simplify √12, we need to find the largest perfect square that divides 12. The perfect squares less than 12 are 1, 4, and 9. The largest perfect square that divides 12 is 4 (12 = 4 * 3). Therefore, we can simplify √12 as:

√12 = √(4 * 3) = √4 * √3 = 2√3

Step 5: Substitute the Simplified Radical Back into the Expression

Now, we substitute the simplified radical 2√3 back into the expression:

-36√12 = -36(2√3)

Step 6: Multiply the Coefficient and the Radical

Finally, we multiply the coefficient -36 by the radical 2√3:

-36(2√3) = -72√3

Therefore, the simplified form of the expression (-9√2)(4√6) is -72√3. This corresponds to option B in the given choices.

Why Other Options Are Incorrect

To further solidify your understanding, let's examine why the other options are incorrect:

  • A. -5√√8: This option is incorrect because it involves taking the square root of a square root, which is not the correct procedure for simplifying the expression. Additionally, the coefficient -5 is incorrect.
  • C. -36√8: This option is incorrect because it simplifies the radicals partially but does not completely simplify √8. As we saw earlier, √8 can be further simplified as 2√2.
  • D. -72√2: This option is incorrect because it simplifies the radicals incorrectly. It seems to have missed the step of combining √2 and √6 under the same radical and then simplifying.

Mastering Radical Simplification: Tips and Tricks

To become proficient in radical simplification, consider these tips and tricks:

  • Memorize Perfect Squares: Knowing the perfect squares (1, 4, 9, 16, 25, 36, etc.) will help you quickly identify factors that can be simplified.
  • Practice Regularly: Consistent practice is key to mastering any mathematical skill. Work through various examples to build your confidence and understanding.
  • Break Down Complex Radicals: If you encounter a radical with a large radicand, try breaking it down into smaller factors. This will make it easier to identify perfect square factors.
  • Use the Product and Quotient Properties: The product and quotient properties of radicals are powerful tools for simplifying expressions. Make sure you understand how to apply them correctly.
  • Check Your Answers: After simplifying an expression, double-check your work to ensure you haven't made any errors.

Conclusion

Simplifying radical expressions is an essential skill in mathematics. By understanding the core concepts and following a step-by-step approach, you can confidently tackle even the most complex expressions. In this article, we successfully simplified the expression (-9√2)(4√6) to -72√3. Remember to practice regularly and apply the tips and tricks discussed to further enhance your understanding of radical simplification.

By mastering radical simplification, you'll not only improve your mathematical skills but also gain a deeper appreciation for the beauty and elegance of mathematics. So, embrace the challenge, practice diligently, and unlock the power of simplified expressions!

This exploration of simplifying the expression (-9√2)(4√6) serves as a stepping stone to more advanced mathematical concepts. As you continue your mathematical journey, remember that every problem is an opportunity to learn and grow. So, keep exploring, keep questioning, and keep simplifying!