Cube Root of 729000

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, ∛729000 is written as 729000^(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 729000, then y³ can be 729000. Since the cube root of 729000 is an exact value, we can write it as 90.

Finding the cube root of a number is to identify the number that must be multiplied three times to result in the target number. Now, we will go through the different ways to find the cube root of 729000. The common methods we follow to find the cube root are given below:

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

Since 729000 is a perfect cube, the prime factorization method is the most straightforward way to find its cube root.

Let's find the cube root of 729000 using the prime factorization method.

First, break down 729000 into its prime factors: 729000 = 2³ × 3⁶ × 5³ × 7³

Since each prime factor has an exponent that is a multiple of 3, we can take the cube root by dividing the exponents by 3:

∛729000 = ∛(2³ × 3⁶ × 5³ × 7³ )

= 2^(3/3) × 3^(6/3) × 5^(3/3) × 7^(3/3)

= 2 × 3^2 × 5 × 7

= 2 × 9 × 5 × 7 = 90

Therefore, the cube root of 729000 is 90.

• 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, 90 × 90 × 90 = 729000, therefore, 729000 is a perfect cube.
   
• Exponent: The exponent form of the number denotes the number of times a number can be multiplied by itself. In 729000^(1/3), ⅓ is the exponent which denotes the cube root of 729000.
   
• Radical sign: The symbol that is used to represent a root is expressed as (∛).
   
• Rational number: The numbers that can be expressed as a fraction or whole number are rational. For example, the cube root of 729000 is rational because it is a whole number (90).