Calculating (0.064)^(2/3) / 1000 A Step By Step Solution
Introduction
In this article, we will delve into the process of calculating the fraction (0.064)^(2/3) / 1000. This seemingly simple expression involves several mathematical concepts, including exponents, fractional powers, and decimal representation. By breaking down the problem step-by-step, we can gain a deeper understanding of how these concepts work together. Our primary goal is to simplify this expression and arrive at a final numerical answer. The beauty of mathematics lies in its precision and the logical steps one can take to solve complex problems. We will leverage the properties of exponents and fractions to simplify this expression, making it easier to compute. This exploration is not just about arriving at the right answer but also about understanding the journey, the mathematical principles involved, and how different concepts interplay. By mastering these fundamentals, we build a solid foundation for tackling more complex mathematical challenges in the future. So, let's embark on this mathematical journey, unravel the intricacies of the expression, and discover the simplified result.
Breaking Down the Problem
To effectively calculate (0.064)^(2/3) / 1000, we'll break the problem down into manageable parts. First, we'll focus on the numerator, (0.064)^(2/3). This part involves a decimal raised to a fractional power, which might seem daunting at first. However, by understanding the properties of exponents and roots, we can simplify this significantly. The key is to recognize that a fractional exponent represents both a power and a root. For instance, x^(a/b) is the same as taking the b-th root of x raised to the power of a. Understanding this fundamental concept is crucial for solving this problem. Next, we'll convert the decimal 0.064 into a more workable form, such as a fraction. This will make it easier to apply the exponent. Converting decimals to fractions is a common technique in mathematics that simplifies calculations, especially when dealing with exponents and roots. Once we've simplified the numerator, we'll move on to the denominator, 1000. This part is relatively straightforward, but it's essential to keep it in mind as we proceed with the calculations. Finally, we'll divide the simplified numerator by the denominator to arrive at our final answer. This step involves applying the rules of fraction division, which is a fundamental arithmetic operation. By breaking down the problem in this way, we make it more approachable and reduce the chances of making errors. Each step builds upon the previous one, leading us towards the solution in a logical and methodical manner.
Simplifying the Numerator: (0.064)^(2/3)
Let's focus on simplifying the numerator, which is (0.064)^(2/3). The first step is to convert the decimal 0.064 into a fraction. We can express 0.064 as 64/1000. This conversion is crucial because it allows us to work with whole numbers and apply the properties of exponents more easily. Fractions provide a clearer representation of the number's value and simplify the subsequent calculations. Now, we have (64/1000)^(2/3). Recall that a fractional exponent represents both a power and a root. Specifically, the exponent 2/3 means we need to take the cube root (the denominator 3) and then raise the result to the power of 2 (the numerator 2). To simplify this further, we can apply the exponent to both the numerator and the denominator separately. This is a key property of exponents: (a/b)^n = a^n / b^n. So, we have (64^(2/3)) / (1000^(2/3)). Now, let's calculate the cube root of 64 and 1000. The cube root of 64 is 4 because 4 * 4 * 4 = 64. The cube root of 1000 is 10 because 10 * 10 * 10 = 1000. So, we now have (4^2) / (10^2). Finally, we calculate 4 squared, which is 16, and 10 squared, which is 100. Thus, the simplified numerator is 16/100. This process of breaking down the fractional exponent and applying the rules of exponents step-by-step allows us to simplify the expression and arrive at a manageable fraction. Understanding these properties is essential for working with exponents and fractional powers in mathematics.
Calculating 64^(2/3)
To further clarify the calculation of 64^(2/3), let's break it down into two steps. As we discussed earlier, the fractional exponent 2/3 indicates that we need to first find the cube root of 64 and then square the result. The cube root of 64, denoted as ∛64, is the number that, when multiplied by itself three times, equals 64. We know that 4 * 4 * 4 = 64, therefore, ∛64 = 4. This step is crucial because it simplifies the base of our exponent. Understanding cube roots is fundamental for working with fractional exponents. Now that we have the cube root, we need to square the result. Squaring a number means multiplying it by itself. So, we need to calculate 4^2, which is 4 * 4. This gives us 16. Therefore, 64^(2/3) = 16. This result is a key component in simplifying the overall expression. By breaking down the fractional exponent into its root and power components, we can perform the calculations more easily. This approach is not only applicable to this specific problem but also to a wide range of mathematical problems involving fractional exponents. Understanding the relationship between roots, powers, and fractional exponents is essential for building a strong foundation in mathematics.
Calculating 1000^(2/3)
Now, let's calculate 1000^(2/3) using the same approach we used for 64^(2/3). The fractional exponent 2/3 tells us to first find the cube root of 1000 and then square the result. The cube root of 1000, denoted as ∛1000, is the number that, when multiplied by itself three times, equals 1000. We know that 10 * 10 * 10 = 1000, therefore, ∛1000 = 10. This step simplifies the base of our exponent, making the next calculation easier. Understanding cube roots is crucial for working with fractional exponents, especially when the base is a large number like 1000. Next, we need to square the result. Squaring a number means multiplying it by itself. So, we need to calculate 10^2, which is 10 * 10. This gives us 100. Therefore, 1000^(2/3) = 100. This result is the denominator of our simplified fraction from the numerator calculation. By following this step-by-step process, we can easily calculate fractional exponents, even when dealing with larger numbers. The key is to break the problem down into smaller, more manageable parts and apply the fundamental principles of exponents and roots. This method is not only effective but also provides a clear understanding of the mathematical concepts involved.
Dividing by 1000
After simplifying the numerator (0.064)^(2/3) to 16/100, we now need to divide this result by the denominator, 1000. The original expression was (0.064)^(2/3) / 1000, and we've simplified the numerator to 16/100. So, we now have (16/100) / 1000. To divide a fraction by a whole number, we can rewrite the whole number as a fraction with a denominator of 1. So, 1000 becomes 1000/1. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1000/1 is 1/1000. Therefore, we need to multiply 16/100 by 1/1000. The multiplication of fractions is straightforward: we multiply the numerators together and the denominators together. So, we have (16 * 1) / (100 * 1000), which equals 16/100000. This fraction can be further simplified by finding the greatest common divisor (GCD) of the numerator and denominator. Both 16 and 100000 are divisible by 16. Dividing both the numerator and the denominator by 16, we get 1/6250. Therefore, the final simplified result of the expression is 1/6250. This step-by-step process of dividing the simplified numerator by the denominator demonstrates the importance of understanding fraction division and simplification. By applying these principles, we can arrive at the final answer in a clear and logical manner.
Final Result and Simplification
After performing all the calculations, we have arrived at the final result: 16/100000. However, it is crucial to simplify this fraction to its lowest terms. Simplification makes the result cleaner and easier to understand. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by the GCD. In this case, the GCD of 16 and 100000 is 16. Both numbers are divisible by 16, which allows us to simplify the fraction significantly. Dividing the numerator 16 by 16 gives us 1. Dividing the denominator 100000 by 16 gives us 6250. Therefore, the simplified fraction is 1/6250. This is the final, simplified answer to our original problem. Converting this fraction to a decimal can also provide a different perspective on the result. Dividing 1 by 6250 gives us 0.00016. This decimal representation is equivalent to the fraction 1/6250 and provides a sense of the magnitude of the result. Throughout this process, we have applied various mathematical principles, including the properties of exponents, fractional powers, roots, fraction division, and simplification. Each step has been carefully executed to ensure accuracy and clarity. The final result, 1/6250 or 0.00016, represents the solution to the expression (0.064)^(2/3) / 1000. This exercise highlights the importance of breaking down complex problems into smaller, manageable steps and applying fundamental mathematical principles to arrive at the solution.
Conclusion
In conclusion, we have successfully calculated and simplified the expression (0.064)^(2/3) / 1000. This problem involved several key mathematical concepts, including fractional exponents, roots, and fraction manipulation. By breaking the problem down into smaller, more manageable steps, we were able to simplify each part and arrive at the final answer. First, we converted the decimal 0.064 into the fraction 64/1000. Then, we tackled the fractional exponent, recognizing that (a/b)^n is equivalent to taking the b-th root of a raised to the power of n. We calculated 64^(2/3) by first finding the cube root of 64, which is 4, and then squaring the result, giving us 16. Similarly, we calculated 1000^(2/3) by finding the cube root of 1000, which is 10, and then squaring the result, giving us 100. This simplified the numerator to 16/100. Next, we divided the simplified numerator by the denominator, 1000. This involved multiplying the fraction 16/100 by the reciprocal of 1000, which is 1/1000. This gave us the fraction 16/100000. Finally, we simplified this fraction by finding the greatest common divisor (GCD) of 16 and 100000, which is 16, and dividing both the numerator and the denominator by 16. This resulted in the simplified fraction 1/6250, which is our final answer. We also converted this fraction to a decimal, which is 0.00016, providing another representation of the result. This exercise demonstrates the power of breaking down complex problems into simpler steps and applying fundamental mathematical principles. By understanding and applying these concepts, we can confidently tackle a wide range of mathematical challenges. The journey from the initial expression to the final simplified answer highlights the elegance and precision of mathematics.
