Cube Root of 59049

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, ∛59049 is written as 59049^(1/3). The cube root is just the opposite operation of finding the cube of a number. For example, assume ‘y’ is the cube root of 59049, then y³ can be 59049. Since the cube root of 59049 is an exact value, it is 39.

Finding the cube root of a number is to identify 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 59049. The common methods we follow to find the cube root are given below:

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

To find the cube root of a non-perfect cube number, we often follow Halley’s method. However, since 59049 is a perfect cube, we can use the prime factorization method.

Let's find the cube root of 59049 using the prime factorization method.

First, express 59049 as a product of its prime factors:

59049 = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3

Group the factors into triples: (3 × 3 × 3) × (3 × 3 × 3) × (3 × 3 × 3)

Take one factor from each group: 3 × 3 × 3 = 27

Thus, the cube root of 59049 is 27.

• 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. A perfect cube always results in a whole number. For example, 3 × 3 × 3 = 27, therefore, 27 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 of a number.

• Radical sign: The symbol that is used to represent a root which is expressed as (∛).

• Rational number: A number that can be expressed as a fraction or a whole number. The cube root of 59049 is rational because it equals 39.