How To Simplify Cube Root Of 875 A Step-by-Step Guide

Simplifying radical expressions is a fundamental concept in mathematics, and it's crucial for anyone studying algebra and beyond. In this comprehensive guide, we will delve into the process of simplifying the cube root of 875, ${\sqrt[3]{875}}$. We'll break down each step, explain the underlying principles, and provide valuable insights to help you master this skill. Whether you're a student looking to ace your math exams or simply someone interested in expanding your mathematical knowledge, this article will equip you with the necessary tools and understanding.

Understanding Radical Expressions

Before diving into the simplification of ${\sqrt[3]{875}}$, let's establish a solid foundation by understanding what radical expressions are and their components. A radical expression consists of a radical symbol (${\sqrt[n]{ }}$), a radicand (the number or expression under the radical), and an index (the small number ${n}$ indicating the type of root). In the expression ${\sqrt[3]{875}}$, the radical symbol is ${\sqrt[3]{ }}$, the radicand is 875, and the index is 3, signifying a cube root. Understanding these components is essential for simplifying any radical expression.

Prime Factorization: The Key to Simplification

The cornerstone of simplifying radicals lies in the prime factorization of the radicand. Prime factorization is the process of breaking down a number into its prime factors, which are numbers divisible only by 1 and themselves (e.g., 2, 3, 5, 7, 11). To simplify ${\sqrt[3]{875}}$, we first need to find the prime factors of 875. Let's break it down:

• 875 is divisible by 5, so 875 = 5 * 175
• 175 is also divisible by 5, so 175 = 5 * 35
• 35 is divisible by 5, so 35 = 5 * 7
• 7 is a prime number

Therefore, the prime factorization of 875 is 5 * 5 * 5 * 7, or ${5^3 * 7}$. This is a crucial step as it allows us to rewrite the radicand in a form that reveals perfect cube factors.

Step-by-Step Simplification of ${\sqrt[3]{875}}$

Now that we have the prime factorization of 875, we can proceed with simplifying the radical expression step by step. This methodical approach ensures accuracy and clarity in the simplification process.

Step 1: Rewrite the Radical Expression

Replace 875 with its prime factorization within the radical:

${\sqrt[3]{875} = \sqrt[3]{5^3 * 7}}$

This step sets the stage for identifying perfect cubes within the radicand. By expressing 875 as a product of its prime factors, we can easily spot the factors that can be extracted from the cube root.

Step 2: Apply the Product Property of Radicals

The product property of radicals states that ${\sqrt[n]{ab} = \sqrt[n]{a} * \sqrt[n]{b}}$. This property allows us to separate the radical into two radicals, each containing a factor from the prime factorization:

${\sqrt[3]{5^3 * 7} = \sqrt[3]{5^3} * \sqrt[3]{7}}$

Applying this property is fundamental for simplifying radicals with multiple factors in the radicand. It isolates the perfect cube (${5^3}$) from the remaining factors.

Step 3: Simplify the Perfect Cube

The cube root of ${5^3}$ is simply 5, since ${5 * 5 * 5 = 125}$ and ${\sqrt[3]{125} = 5}$. Therefore:

${\sqrt[3]{5^3} * \sqrt[3]{7} = 5 * \sqrt[3]{7}}$

This step is where the actual simplification occurs. By extracting the cube root of the perfect cube factor, we reduce the radical expression to its simplest form. This transformation is the essence of simplifying radicals.

Step 4: Final Simplified Form

The simplified form of ${\sqrt[3]{875}}$ is ${5\sqrt[3]{7}}$. This means that the cube root of 875 can be expressed as 5 times the cube root of 7. The radical ${\sqrt[3]{7}}$ cannot be simplified further because 7 is a prime number and does not have any perfect cube factors. This final answer represents the most concise and simplified form of the original expression.

Common Mistakes to Avoid

When simplifying radicals, it's easy to make common mistakes that can lead to incorrect answers. Being aware of these pitfalls is crucial for achieving accuracy. Here are some frequent errors to watch out for:

Mistake 1: Incorrect Prime Factorization

A flawed prime factorization will inevitably lead to an incorrect simplification. Always double-check your prime factors to ensure they are accurate. For instance, if you incorrectly factor 875, you might not identify the perfect cube factor of ${5^3}$.

Mistake 2: Forgetting the Index

It's important to remember the index of the radical. For example, ${\sqrt[3]{ }}$ represents a cube root, while ${\sqrt{ }}$ (without an index) represents a square root. Confusing these can lead to incorrect simplification. Always pay close attention to the index when applying the properties of radicals.

Mistake 3: Incorrectly Applying the Product Property

The product property of radicals is powerful, but it must be applied correctly. Ensure that you are separating the radicand into factors and applying the radical to each factor individually. A common error is to attempt to simplify terms within the radical without first separating them.

Mistake 4: Not Simplifying Completely

A radical expression is only fully simplified when there are no more perfect nth roots (where n is the index) within the radicand. Always check if the remaining radicand can be further simplified. For example, if you stopped at ${\sqrt[3]{5^3 * 7}}$ without extracting the cube root of ${5^3}$, you wouldn't have fully simplified the expression.

Practice Problems

To solidify your understanding of simplifying radicals, practice is key. Here are a few additional problems for you to try:

1. ${\sqrt[3]{16}}$
2. ${\sqrt[3]{54}}$
3. ${\sqrt[3]{128}}$

By working through these problems, you'll reinforce your skills and gain confidence in simplifying radical expressions. Remember to follow the same steps we outlined for ${\sqrt[3]{875}}$: prime factorization, applying the product property, and simplifying perfect cubes.

Conclusion

Simplifying radical expressions like ${\sqrt[3]{875}}$ is an essential skill in mathematics. By understanding the principles of prime factorization, the product property of radicals, and avoiding common mistakes, you can confidently tackle these types of problems. Remember, the key to success lies in a methodical approach and consistent practice. This comprehensive guide has provided you with the knowledge and steps needed to simplify cube roots effectively. Keep practicing, and you'll master this valuable skill in no time!

By following these steps and understanding the underlying concepts, you can confidently simplify radical expressions. Remember to practice regularly to reinforce your skills and build your mathematical fluency. Mastering these techniques will be invaluable as you progress in your mathematical journey. Happy simplifying!