Simplify The Cube Root Of 216/343 A Step-by-Step Guide

by THE IDEN 55 views

In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to express complex equations in their most basic form. This process not only makes the expressions easier to understand but also facilitates further calculations and manipulations. In this comprehensive guide, we will delve into the intricacies of simplifying the cube root of a fraction, specifically $\sqrt[3]{\frac{216}{343}}$. We will embark on a step-by-step journey, exploring the underlying principles and techniques involved in this simplification process. Our exploration will encompass the concepts of prime factorization, cube roots, and the properties of radicals, providing a holistic understanding of the simplification process.

At the heart of simplifying radical expressions lies the concept of cube roots. A cube root of a number is a value that, when multiplied by itself three times, yields the original number. Mathematically, if we have a number 'x', its cube root is denoted as $\sqrt[3]{x}$, and it satisfies the equation $(\sqrt[3]{x})^3 = x$. For instance, the cube root of 8 is 2, because 2 multiplied by itself three times (2 * 2 * 2) equals 8. Similarly, the cube root of 27 is 3, as 3 * 3 * 3 equals 27. Understanding cube roots is crucial for simplifying expressions involving radicals.

Before we can effectively simplify the cube root of a fraction, we need to master the art of prime factorization. Prime factorization is the process of breaking down a composite number into its prime factors – the prime numbers that, when multiplied together, give the original number. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and so on. For example, the prime factorization of 12 is 2 * 2 * 3, as 2 and 3 are prime numbers and their product is 12. Prime factorization is a fundamental tool in simplifying radicals, as it allows us to identify perfect cube factors within the radicand.

Now, let's apply our knowledge of cube roots and prime factorization to simplify the expression $\sqrt[3]{\frac{216}{343}}$. This expression represents the cube root of the fraction 216/343. To simplify this, we will simplify the cube root of the numerator and the cube root of the denominator separately.

Step 1: Prime Factorization of 216

First, we find the prime factorization of 216. We can start by dividing 216 by the smallest prime number, 2:

  • 216 ÷ 2 = 108
  • 108 ÷ 2 = 54
  • 54 ÷ 2 = 27
  • 27 ÷ 3 = 9
  • 9 ÷ 3 = 3
  • 3 ÷ 3 = 1

Thus, the prime factorization of 216 is 2 * 2 * 2 * 3 * 3 * 3, which can be written as $2^3 * 3^3$. This reveals that 216 is a perfect cube, as it can be expressed as the product of cubes.

Step 2: Prime Factorization of 343

Next, we find the prime factorization of 343. Since 343 is not divisible by 2, we try the next prime number, 3. It's not divisible by 3 either, so we move on to 5. 343 is not divisible by 5. However, it is divisible by 7:

  • 343 ÷ 7 = 49
  • 49 ÷ 7 = 7
  • 7 ÷ 7 = 1

Thus, the prime factorization of 343 is 7 * 7 * 7, which can be written as $7^3$. This indicates that 343 is also a perfect cube.

Step 3: Applying Cube Root Properties

Now that we have the prime factorizations of both 216 and 343, we can rewrite the original expression as:

2163433=23∗33733\sqrt[3]{\frac{216}{343}} = \sqrt[3]{\frac{2^3 * 3^3}{7^3}}

Using the property of radicals that states $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$, we can separate the cube root of the fraction into the cube root of the numerator divided by the cube root of the denominator:

23∗33733=23∗333733\sqrt[3]{\frac{2^3 * 3^3}{7^3}} = \frac{\sqrt[3]{2^3 * 3^3}}{\sqrt[3]{7^3}}

We also have the property $\sqrt[n]{a * b} = \sqrt[n]{a} * \sqrt[n]{b}$, which allows us to separate the cube root of the product in the numerator:

23∗333733=233∗333733\frac{\sqrt[3]{2^3 * 3^3}}{\sqrt[3]{7^3}} = \frac{\sqrt[3]{2^3} * \sqrt[3]{3^3}}{\sqrt[3]{7^3}}

Finally, using the definition of cube roots, we know that $\sqrt[3]{x^3} = x$, so we can simplify the expression further:

233∗333733=2∗37\frac{\sqrt[3]{2^3} * \sqrt[3]{3^3}}{\sqrt[3]{7^3}} = \frac{2 * 3}{7}

Step 4: Final Simplification

Now, we simply multiply the numbers in the numerator:

2∗37=67\frac{2 * 3}{7} = \frac{6}{7}

Therefore, the simplified form of $\sqrt[3]{\frac{216}{343}}$ is $\frac{6}{7}$.

In this comprehensive guide, we have successfully simplified the cube root of the fraction 216/343. Our journey involved understanding the concept of cube roots, mastering prime factorization, and applying the properties of radicals. By breaking down the problem into manageable steps, we were able to simplify the expression and arrive at the final answer of 6/7. This process exemplifies the power of mathematical principles in simplifying complex expressions and making them more accessible. Simplifying radicals is not just a mathematical exercise; it's a skill that enhances problem-solving abilities and provides a deeper understanding of mathematical concepts. The ability to simplify expressions is a cornerstone of mathematical proficiency, allowing for more efficient calculations and a clearer grasp of mathematical relationships. By mastering the techniques outlined in this guide, you will be well-equipped to tackle a wide range of simplification problems, furthering your mathematical journey and fostering a deeper appreciation for the elegance and power of mathematics. Remember, practice is key to mastering any mathematical skill, so continue to explore and challenge yourself with new problems.

Cube roots, prime factorization, simplifying radicals, mathematical expressions, fractions, perfect cubes, radicals properties, numerator, denominator, mathematical simplification.