Simplify Cube Root Of 216: A Step-by-Step Guide

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In the realm of mathematics, simplifying radical expressions is a fundamental skill. Among these, cube roots hold a special place, often presenting a unique challenge. This article delves into the process of simplifying the expression 5 ⋅ ³√216, providing a step-by-step guide and exploring the underlying concepts.

Understanding Cube Roots

Before we tackle the main problem, let's solidify our understanding of cube roots. A cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2 because 2 ⋅ 2 ⋅ 2 = 8. The symbol for the cube root is ³√, which distinguishes it from the square root symbol √.

Key Concepts

  • Perfect Cubes: Numbers that can be obtained by cubing an integer (e.g., 1, 8, 27, 64, 125, 216) are called perfect cubes. Recognizing perfect cubes is crucial for simplifying cube roots.
  • Prime Factorization: Breaking down a number into its prime factors (prime numbers that multiply together to give the original number) is a powerful technique for simplifying radicals.
  • Simplifying Radicals: The general principle is to extract any perfect cube factors from under the radical sign. This involves expressing the radicand (the number under the radical) as a product of a perfect cube and another factor.

Step-by-Step Solution: 5 ⋅ ³√216

Now, let's apply these concepts to simplify the given expression, 5 ⋅ ³√216. The core of the problem lies in simplifying the cube root of 216. Here’s a detailed breakdown:

1. Identify the Radicand

The radicand in this expression is 216. Our goal is to determine if 216 is a perfect cube or if it contains any perfect cube factors.

2. Prime Factorization of 216

To find the perfect cube factors, we'll break down 216 into its prime factors. This involves finding the prime numbers that multiply together to give 216. Here’s the prime factorization:

  • 216 = 2 ⋅ 108
  • 108 = 2 ⋅ 54
  • 54 = 2 ⋅ 27
  • 27 = 3 ⋅ 9
  • 9 = 3 ⋅ 3

Combining these, we get the prime factorization of 216 as 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3, which can be written as 2³ ⋅ 3³.

3. Express as a Product of Perfect Cubes

From the prime factorization, we can see that 216 can be expressed as the product of two perfect cubes: 2³ (which is 8) and 3³ (which is 27). Therefore, 216 = 2³ ⋅ 3³ = 8 ⋅ 27.

4. Apply the Cube Root

Now we can rewrite the cube root of 216 using the perfect cube factors:

³√216 = ³√(2³ ⋅ 3³)

Using the property that the cube root of a product is the product of the cube roots (³√(a ⋅ b) = ³√a ⋅ ³√b), we get:

³√(2³ ⋅ 3³) = ³√2³ ⋅ ³√3³

The cube root of a number cubed is simply the number itself. Therefore:

³√2³ = 2

³√3³ = 3

So, ³√216 = 2 ⋅ 3 = 6.

5. Substitute Back into the Original Expression

Now that we've simplified ³√216 to 6, we can substitute it back into the original expression:

5 ⋅ ³√216 = 5 ⋅ 6

6. Final Simplification

Finally, we perform the multiplication:

5 ⋅ 6 = 30

Thus, the simplified form of 5 ⋅ ³√216 is 30.

Alternative Approach: Recognizing Perfect Cubes

An alternative method to simplify ³√216 is by recognizing that 216 is a perfect cube. If you know your perfect cubes, you might recall that 6³ = 6 ⋅ 6 ⋅ 6 = 216. Therefore, ³√216 = 6, and the rest of the steps follow as before.

Common Mistakes to Avoid

When simplifying cube roots, several common mistakes can occur. Being aware of these pitfalls can help prevent errors:

  • Confusing Cube Roots with Square Roots: Ensure you are taking the cube root and not the square root. The cube root requires a factor to appear three times, while the square root requires a factor to appear twice.
  • Incorrect Prime Factorization: Double-check your prime factorization to ensure accuracy. An incorrect factorization will lead to an incorrect simplification.
  • Forgetting to Multiply the Coefficient: After simplifying the cube root, remember to multiply the result by any coefficient that was originally present (in this case, 5).
  • Not Fully Simplifying: Always ensure that the radicand has no remaining perfect cube factors. Continue simplifying until this is the case.

Examples

To further illustrate the concept, let's work through a few more examples of simplifying cube root expressions.

Example 1: Simplify 2 ⋅ ³√54

  1. Prime Factorization of 54:
    • 54 = 2 ⋅ 27
    • 27 = 3 ⋅ 9
    • 9 = 3 ⋅ 3
    • So, 54 = 2 ⋅ 3 ⋅ 3 ⋅ 3 = 2 ⋅ 3³
  2. Rewrite the Expression:
    • 2 ⋅ ³√54 = 2 ⋅ ³√(2 ⋅ 3³)
  3. Apply the Cube Root Property:
    • 2 ⋅ ³√(2 ⋅ 3³) = 2 ⋅ ³√2 ⋅ ³√3³
  4. Simplify the Cube Root:
    • ³√3³ = 3
  5. Final Simplification:
    • 2 ⋅ ³√2 ⋅ 3 = 6 ⋅ ³√2

Thus, 2 ⋅ ³√54 simplifies to 6 ⋅ ³√2.

Example 2: Simplify -4 ⋅ ³√128

  1. Prime Factorization of 128:
    • 128 = 2 ⋅ 64
    • 64 = 2 ⋅ 32
    • 32 = 2 ⋅ 16
    • 16 = 2 ⋅ 8
    • 8 = 2 ⋅ 4
    • 4 = 2 ⋅ 2
    • So, 128 = 2⁷ = 2³ ⋅ 2³ ⋅ 2
  2. Rewrite the Expression:
    • -4 ⋅ ³√128 = -4 ⋅ ³√(2³ ⋅ 2³ ⋅ 2)
  3. Apply the Cube Root Property:
    • -4 ⋅ ³√(2³ ⋅ 2³ ⋅ 2) = -4 ⋅ ³√2³ ⋅ ³√2³ ⋅ ³√2
  4. Simplify the Cube Root:
    • ³√2³ = 2
  5. Final Simplification:
    • -4 ⋅ 2 ⋅ 2 ⋅ ³√2 = -16 ⋅ ³√2

Thus, -4 ⋅ ³√128 simplifies to -16 ⋅ ³√2.

Practical Applications of Simplifying Cube Roots

Simplifying cube roots is not just an abstract mathematical exercise; it has practical applications in various fields. Here are a few examples:

  • Engineering: Engineers often encounter cube roots when calculating volumes, especially when dealing with three-dimensional shapes. Simplifying these expressions can make calculations more manageable.
  • Physics: In physics, cube roots appear in formulas related to volume, density, and other physical properties. Simplifying these expressions can aid in problem-solving.
  • Computer Graphics: Cube roots are used in algorithms for 3D graphics and rendering. Simplifying these expressions can improve the efficiency of these algorithms.
  • Mathematics: Simplifying radicals is a fundamental skill in algebra and calculus. It's essential for solving equations, simplifying expressions, and performing other mathematical operations.

Conclusion

Simplifying expressions involving cube roots is a valuable skill in mathematics. By understanding the concept of cube roots, prime factorization, and perfect cubes, we can efficiently simplify complex expressions. The step-by-step method outlined in this article provides a clear approach to simplifying expressions like 5 ⋅ ³√216. By recognizing perfect cubes and applying the properties of radicals, we can simplify cube root expressions with confidence. Remember to practice and apply these techniques to master the art of simplifying cube roots. The ability to simplify such expressions not only enhances your mathematical proficiency but also prepares you for more advanced concepts and practical applications in various fields.

By consistently applying these principles, you’ll find that simplifying cube roots becomes second nature, enhancing your mathematical problem-solving skills and preparing you for more advanced concepts in mathematics and related fields.