Evaluate Cube Root Of 16/250: A Step-by-Step Guide

Are you struggling with simplifying radical expressions? Do cube roots seem daunting? Fear not! This comprehensive guide will walk you through the process of evaluating the expression $\sqrt[3]{\frac{16}{250}}$, breaking down each step in a clear and understandable manner. We'll cover everything from simplifying fractions to understanding cube roots, ensuring you grasp the underlying concepts. Let's dive in and conquer this mathematical challenge together! This article aims to provide a detailed, step-by-step solution on how to evaluate the cube root of the fraction 16/250. Understanding how to simplify such expressions is crucial in mathematics, especially in algebra and calculus. Simplifying radical expressions like this involves breaking down the numbers into their prime factors and then applying the properties of radicals. The primary goal here is to express the fraction inside the cube root in its simplest form before attempting to find the cube root. This involves identifying common factors in the numerator and the denominator and canceling them out. By understanding these initial steps, you can make the entire process more manageable and less prone to errors. So, let’s begin by understanding the initial setup and the fundamental principles we will be applying.

1. Simplify the Fraction Inside the Cube Root

The first crucial step in evaluating the cube root is to simplify the fraction inside the radical. We have $\sqrt[3]{\frac{16}{250}}$. To simplify the fraction $\frac{16}{250}$, we need to find the greatest common divisor (GCD) of 16 and 250. Both numbers are even, so we can start by dividing both by 2. Dividing 16 by 2 gives us 8, and dividing 250 by 2 gives us 125. So, the fraction becomes $\frac{8}{125}$. Now, we need to check if 8 and 125 have any common factors other than 1. The factors of 8 are 1, 2, 4, and 8. The factors of 125 are 1, 5, 25, and 125. The only common factor is 1, which means the fraction $\frac{8}{125}$ is in its simplest form. By simplifying the fraction, we've made the expression much easier to work with. This simplification not only reduces the complexity of the numbers but also sets us up nicely for the next step, which involves recognizing perfect cubes. When dealing with cube roots, identifying perfect cubes is essential for simplifying the expression effectively. The simplified fraction now allows us to clearly see if the numerator and denominator are perfect cubes, making the subsequent steps more straightforward and less confusing. Now that we have the fraction in its simplest form, we can proceed to the next step of evaluating the cube root.

2. Recognize Perfect Cubes

Now that we have the simplified fraction $\frac{8}{125}$, the next step is to recognize if the numerator and the denominator are perfect cubes. A perfect cube is a number that can be obtained by cubing an integer (raising it to the power of 3). Let's examine the numerator, 8. We need to find a number that, when multiplied by itself three times, equals 8. We know that $2 \times 2 \times 2 = 2^3 = 8$. So, 8 is indeed a perfect cube. Next, let's look at the denominator, 125. We need to find a number that, when cubed, equals 125. We can try a few numbers: $3^3 = 27$, $4^3 = 64$, and $5^3 = 125$. So, 125 is also a perfect cube, as $5^3 = 125$. Recognizing these perfect cubes is a critical step because it allows us to simplify the cube root expression. If the numbers were not perfect cubes, we would need to look for other simplification techniques, such as prime factorization or estimation. However, in this case, because both 8 and 125 are perfect cubes, we can directly take their cube roots. Identifying perfect cubes makes the evaluation of the cube root much more straightforward. It transforms the problem from a potentially complex calculation to a simple extraction of the cube root. This step highlights the importance of knowing common perfect cubes, which can significantly speed up the process of simplifying radical expressions.

3. Apply the Cube Root

Having identified that 8 and 125 are perfect cubes, we can now apply the cube root to both the numerator and the denominator separately. Recall that $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$. Applying this property to our expression, we get: $\sqrt[3]{\frac{8}{125}} = \frac{\sqrt[3]{8}}{\sqrt[3]{125}}$. Now, we need to find the cube root of 8 and the cube root of 125. We already determined that $2^3 = 8$, so $\sqrt[3]{8} = 2$. Similarly, we found that $5^3 = 125$, so $\sqrt[3]{125} = 5$. Substituting these values back into our expression, we get: $\frac{\sqrt[3]{8}}{\sqrt[3]{125}} = \frac{2}{5}$. Therefore, the cube root of $\frac{16}{250}$, simplified, is $\frac{2}{5}$. Applying the cube root to the numerator and denominator separately is a fundamental step in simplifying radical expressions involving fractions. This approach breaks down the problem into smaller, more manageable parts. By finding the cube roots of the individual components, we can easily arrive at the final simplified form. This step reinforces the importance of understanding the properties of radicals and how they can be used to simplify complex expressions. It also demonstrates the elegance of mathematical operations when applied correctly.

4. State the Final Answer

After applying the cube root and simplifying, we have arrived at the final answer. The expression $\sqrt[3]{\frac{16}{250}}$ simplifies to $\frac{2}{5}$. This result is a simplified fraction, representing the exact value of the original expression. To recap the steps we took: We first simplified the fraction inside the cube root from $\frac{16}{250}$ to $\frac{8}{125}$. Then, we recognized that both 8 and 125 are perfect cubes. We applied the cube root to the numerator and the denominator separately, finding that $\sqrt[3]{8} = 2$ and $\sqrt[3]{125} = 5$. Finally, we expressed the result as the fraction $\frac{2}{5}$. Stating the final answer clearly is essential in mathematics to ensure the solution is communicated effectively. In this case, the simplified form $\frac{2}{5}$ is much easier to understand and work with than the original expression. This process highlights the importance of simplification in mathematics, as it often leads to more manageable and understandable results. By clearly stating the final answer, we complete the problem-solving process and provide a concise and accurate solution. This final step is crucial for ensuring clarity and completeness in mathematical problem-solving.

Conclusion

In this guide, we have thoroughly explored how to evaluate the expression $\sqrt[3]{\frac{16}{250}}$. We began by simplifying the fraction inside the cube root, then recognized and applied the cube root to both the numerator and the denominator. The key steps included simplifying fractions, recognizing perfect cubes, and applying the properties of radicals. By following these steps, we successfully simplified the expression to its final form, $\frac{2}{5}$. Understanding these techniques is crucial for tackling more complex problems involving radicals and cube roots. The ability to simplify expressions not only makes calculations easier but also enhances your overall understanding of mathematical concepts. This comprehensive approach to problem-solving is a valuable skill that can be applied in various mathematical contexts. We hope this step-by-step guide has provided you with a clear and concise understanding of how to evaluate cube roots of fractions. Keep practicing, and you'll master these skills in no time! For further learning and practice on similar topics, you might find helpful resources on websites like Khan Academy's Algebra section.