**Introduction**

Understanding mathematical concepts such as cube roots is fundamental in both academic and real-world contexts. The cube root of a number is a value that, when multiplied by itself twice (i.e., raised to the power of three), yields the original number. In this article, we will delve deeply into the concept of cube roots, specifically focusing on finding the cube root of 64. We will explore the mathematical process, real-world applications, and related concepts to provide a comprehensive understanding.

**Definition of Cube Root**

**Mathematical Explanation**

The **cube root** of a number xxx is a number yyy such that: y3=xy^3 = xy3=x

In other words, the cube root of xxx is the value that, when raised to the power of three, results in xxx. This can be mathematically represented as: x3=y\sqrt[3]{x} = y3x=y

**Example with 64**

To find the cube root of 64, we need to determine a number that, when multiplied by itself three times, equals 64. Mathematically, this can be expressed as: 643=y\sqrt[3]{64} = y364=y where y3=64y^3 = 64y3=64

**Calculating the Cube Root of 64**

**Manual Calculation**

Let’s determine the cube root of 64 step by step:

**List the factors**: Identify the prime factors of 64. 64=2×2×2×2×2×2=2664 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^664=2×2×2×2×2×2=26**Group the factors into triples**: Since we are finding the cube root, we group the factors into sets of three. 64=(2×2×2)×(2×2×2)=(23)×(23)64 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) = (2^3) \times (2^3)64=(2×2×2)×(2×2×2)=(23)×(23)**Simplify the expression**: Each group of three identical factors can be taken out of the cube root. 643=(23)×(23)3=2×2=4\sqrt[3]{64} = \sqrt[3]{(2^3) \times (2^3)} = 2 \times 2 = 4364=3(23)×(23)=2×2=4

Thus, the cube root of 64 is: 643=4\sqrt[3]{64} = 4364=4

**Verification**

To verify, we can cube the result: 43=4×4×4=644^3 = 4 \times 4 \times 4 = 6443=4×4×4=64

This confirms that our calculation is correct. Therefore, the cube root of 64 is indeed 4.

**Applications of Cube Roots**

**Real-World Applications**

Understanding cube roots is essential in various fields, including:

**Geometry**: Calculating the dimensions of cubes when given the volume.- Example: If a cube has a volume of 64 cubic units, each side of the cube will be the cube root of 64, which is 4 units.
**Physics**: Understanding the relationship between volume and side length in three-dimensional spaces.- Example: In determining the side length of a cube-shaped container that can hold 64 cubic meters of liquid, the side length is the cube root of the volume.
**Engineering**: Designing and analyzing structures and materials that involve cubic measurements.

**Mathematical Applications**

**Solving Equations**: Cube roots are used in algebra to solve cubic equations.- Example: In the equation x3=64x^3 = 64×3=64, finding xxx involves calculating the cube root of 64, resulting in x=4x = 4x=4.
**Data Analysis**: In statistics, cube roots can be used to normalize data that is exponentially distributed.

**Related Concepts**

**Higher Order Roots**

**Fourth Root**: The fourth root of a number xxx is a number yyy such that y4=xy^4 = xy4=x.**N-th Root**: Generalizing, the n-th root of a number xxx is a number yyy such that yn=xy^n = xyn=x.

**Cube Root Functions**

**Function Definition**: The cube root function f(x)=x3f(x) = \sqrt[3]{x}f(x)=3x maps any real number xxx to its cube root.**Graphical Representation**: The graph of the cube root function is a curve that passes through the origin and extends infinitely in both directions.

**Conclusion**

The cube root of 64 is 4, a result derived through mathematical calculation and verification. Understanding cube roots is vital for various applications in geometry, physics, engineering, and beyond. By mastering the concept of cube roots and their calculations, one gains a powerful tool for solving a wide range of practical and theoretical problems.