Cube Root of 32768

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, ∛32768 is written as 32768^(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 32768, then y³ = 32768. Since the cube root of 32768 is an exact value, we can find it precisely as 32.

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 32768. The common methods we follow to find the cube root are given below:

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

Since 32768 is a perfect cube, the prime factorization method is efficient for finding its cube root.

Let's find the cube root of 32768 using the prime factorization method.

First, express 32768 as a product of its prime factors.

32768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹⁵

To find the cube root, group the prime factors in triples: (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

Each group of (2 × 2 × 2) is 8, and there are five such groups,

so: ∛(2¹⁵) = 2^(15/3) = 2⁵ = 32

Therefore, the cube root of 32768 is 32.

• 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: 2 × 2 × 2 = 8, therefore, 8 is a perfect cube.
   
• Exponent: The exponent form of the number denotes the number of times a number can be multiplied by itself. In 32768^(1/3), ⅓ is the exponent which denotes the cube root of 32768.
   
• 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 ratio of two integers. For example, the cube root of 32768 is rational because it is 32, a whole number.