Cube Root of 421875

We have learned the definition of the cube root. Now, let’s learn how it is represented using a symbol and exponent. The symbol we use to express the cube root is the radical sign (∛), and the exponent we use is ⅓.

In exponential form, ∛421875 is written as \(421875^{1/3}\). The cube root is just the opposite operation of finding the cube of a number. For example, assume ‘y’ as the cube root of 421875, then \(y³ = 421875\). The cube root of 421875 is an exact whole number, which is 75.

Finding the cube root of a number means identifying the number that must be multiplied three times resulting in the target number. Now, we will go through the different ways to find the cube root of 421875. The common methods we follow to find the cube root are given below: 

• Prime factorization method 
• Approximation method 
• Subtraction method
• Halley’s method

Since 421875 is a perfect cube, we can use the prime factorization method to find its cube root.

Let's find the cube root of 421875 using the prime factorization method:

1. First, we find the prime factors of 421875: 421875 = 3 × 3 × 3 × 5 × 5 × 5 × 5 × 5 × 5

2. Group the prime factors into triples: (3 × 3 × 3) and (5 × 5 × 5) and (5 × 5 × 5)

3. Take one factor from each group: ∛421875 = 3 × 5 × 5 = 75

Therefore, the cube root of 421875 is 75.

• Cube root: The number that is multiplied three times by itself to get the given number is the cube root of that number.
   
• Perfect cube: A number is a perfect cube when it is the product of multiplying a number three times by itself. For example: 5 × 5 × 5 = 125, therefore, 125 is a perfect cube.
   
• Exponent: The exponent form of the number denotes the number of times a number can be multiplied by itself. In \(a^{1/3}\), ⅓ is the exponent which denotes the cube root.
   
• Prime factorization: The process of breaking down a number into its prime factors to simplify calculations like finding roots.
   
• Radical sign: The symbol (∛) used to represent the cube root of a number.