Simplified Form Of √64x¹⁶ Explained

Navigating the realm of mathematics often involves simplifying complex expressions into their most concise forms. This process, known as simplification, not only makes expressions easier to understand but also facilitates further calculations and problem-solving. In this comprehensive guide, we will delve into the simplification of the algebraic expression √64x¹⁶, exploring the underlying mathematical principles and techniques involved. Our journey will culminate in identifying the correct simplified form from a set of multiple-choice options, providing you with a solid understanding of this fundamental mathematical concept.

Understanding the Basics: Square Roots and Exponents

Before we embark on the simplification process, it is crucial to establish a firm grasp of the fundamental concepts involved: square roots and exponents. These concepts serve as the building blocks for simplifying expressions like √64x¹⁶.

Square Roots: Unveiling the Root

The square root of a number is a value that, when multiplied by itself, yields the original number. In mathematical notation, the square root of a number 'a' is represented as √a. For instance, the square root of 9 (√9) is 3 because 3 multiplied by itself (3 * 3) equals 9. Similarly, the square root of 25 (√25) is 5, as 5 * 5 = 25.

The concept of square roots extends to algebraic expressions as well. The square root of an expression involving variables is a term that, when multiplied by itself, results in the original expression. For example, the square root of x² (√x²) is x because x multiplied by itself (x * x) equals x². This principle holds true for more complex expressions as well.

Exponents: Expressing Repeated Multiplication

Exponents provide a concise way to represent repeated multiplication of a number or variable by itself. In the expression aⁿ, 'a' is the base and 'n' is the exponent. The exponent indicates the number of times the base is multiplied by itself. For instance, in the expression 2³, 2 is the base and 3 is the exponent. This signifies that 2 is multiplied by itself three times (2 * 2 * 2), resulting in 8.

Exponents play a crucial role in simplifying expressions involving square roots. When taking the square root of an expression with an exponent, we essentially divide the exponent by 2. For example, the square root of x⁴ (√x⁴) is x² because 4 divided by 2 equals 2. This principle stems from the fact that taking the square root is the inverse operation of squaring, and squaring involves multiplying exponents by 2.

Deconstructing the Expression √64x¹⁶

Now that we have solidified our understanding of square roots and exponents, let's turn our attention to the expression at hand: √64x¹⁶. To simplify this expression, we will break it down into its individual components and apply the principles we have learned.

The expression √64x¹⁶ can be viewed as the square root of the product of two terms: 64 and x¹⁶. In mathematical notation, this can be represented as √(64 * x¹⁶). To simplify this expression, we can take the square root of each term individually and then multiply the results.

Simplifying the Numerical Term: √64

The first term we will address is 64. We need to find the square root of 64, which is the number that, when multiplied by itself, equals 64. From our knowledge of basic squares, we know that 8 * 8 = 64. Therefore, the square root of 64 (√64) is 8.

Simplifying the Variable Term: √x¹⁶

Next, we will simplify the variable term, x¹⁶. As we discussed earlier, taking the square root of an expression with an exponent involves dividing the exponent by 2. In this case, the exponent of x is 16. Dividing 16 by 2, we get 8. Therefore, the square root of x¹⁶ (√x¹⁶) is x⁸.

Combining the Simplified Terms

Having simplified both the numerical and variable terms, we can now combine them to obtain the simplified form of the original expression. We found that √64 = 8 and √x¹⁶ = x⁸. Therefore, the simplified form of √64x¹⁶ is 8 * x⁸, which is commonly written as 8x⁸.

Identifying the Correct Option

Now that we have simplified the expression √64x¹⁶ to 8x⁸, let's examine the multiple-choice options provided and identify the correct answer:

A. 8x⁴ B. 8x⁸ C. 32x⁴ D. 32x⁸

By comparing our simplified result, 8x⁸, with the given options, we can clearly see that option B, 8x⁸, matches our solution. Therefore, the correct answer is B.

Conclusion: Mastering the Art of Simplification

In this comprehensive guide, we have successfully simplified the algebraic expression √64x¹⁶, demonstrating the power of mathematical principles and techniques in transforming complex expressions into their most concise forms. We began by establishing a firm understanding of square roots and exponents, the fundamental building blocks of this simplification process. We then deconstructed the expression, simplifying each term individually before combining them to arrive at the final answer.

Through this journey, we have not only identified the correct simplified form of √64x¹⁶ but also gained a deeper appreciation for the art of mathematical simplification. This skill is invaluable in various mathematical contexts, enabling us to solve problems more efficiently and effectively. By mastering the principles and techniques discussed in this guide, you will be well-equipped to tackle a wide range of simplification challenges in your mathematical endeavors.

Remember, the key to success in mathematics lies in a solid understanding of the fundamentals and the ability to apply them creatively. So, embrace the challenge, practice diligently, and watch your mathematical prowess soar.