Cube Root by Factorization Method

Cube and cube root is one of the most interesting concepts in Mathematics. Whenever a number (x) is multiplied three times, then the resultant number is known as the cube of that number. The cube of a number can also be exponentially represented as the number to the power of 3. The cube root of any real number is that number which when raised to the power of 3 gives the answer equal to the number whose cube root is to be determined. The cube root of a number can also be exponentially represented as the number to the power 1/3.

Cube root is the factor of a number that is multiplied by itself three times to get the resultant number.

Cube root is represented by the symbol \[ \sqrt[3]{ } \].

For example,

4×4×4= 4³ = 64

So,\[ \sqrt[3]{ 64} \] = 4.

The cube root of a number ‘x’ is denoted as \[ \sqrt[3]{x} \] or (x)⅓. The cube of natural numbers is called the perfect cube numbers. Any perfect cube number will have a cube root equal to a whole number. Cube root of a number can be found either by estimation method or by prime factorization method. However, these two methods are valid only for perfect cube numbers. 

Properties of the Cube

• Cubes of odd numbers are odd.

• Cubes of even numbers are even.

• Cubes of numbers that end with 2 will end in 8. Similarly, cubes of numbers that end with 8 will always end with 2.

• Cubes of numbers that end with 3 will end in 7. Similarly, cubes of numbers that end with 7 will always end with 3.

Finding Cube Root of a Number by Prime Factorization Method

Cube root of a number which is a perfect cube can be determined by the prime factorization method. The name prime factorization method is because the method involves the process of resolution of the number whose cube root is to be found into its prime factors. The steps to be followed in order to find the cube root of a number using the prime factorization method are summarised below with an example. 

Step1:

Obtain the number whose cube root is to be found. 

Example: Let us consider a perfect cube number 32768 whose cube root is to be determined.

Step 2:

Start dividing the number by the lowest possible prime number until it is not completely divisible by that prime number. Once the number is not divisible by the lowest prime number assumed, try with the next higher prime number. Continue the division process till the final number obtained as a quotient is also a prime number. 

Example:

Step 3:

Express the number whose square root is to be determined as the product of their primes.

Example:

32768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Step 4: 

Every three identical factors are put in groups.

Example:

32768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Step 5:

Take one element from each group and find the product. The product thus obtained is equal to the cube root of the number.

Example:

\[ \sqrt[3]{ 32768} \] = 2 x 2 x 2 x 2 x 2

\[ \sqrt[3]{ 32768} \] = 32

Cube Root by Factorization Method Example Problems

1. Find the cube root of 46656 using the prime factorization method.

Solution:

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46656 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3 

\[ \sqrt[3]{ 46656} \] = 2×2×3×3 =36

2. Find the smallest number by which 243 should be multiplied to get a perfect cube number.

Solution:

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243 = 3 x 3 x 3 x 3 x 3

243 should be multiplied by 3 to make it a perfect cube number.

Fun Facts about Cube Roots

• Cube root of the numbers ending with 1, 8, 7, 4, 5, 6, 3, 2, 9, and 0 may have 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, respectively, in their unit’s place.

• The cube root of a non perfect cube number cannot be determined by the prime factorization method. 

• The number of digits in a number and its cube root are listed below:

Conclusion

The cube of a number (x) is the resultant number when it is multiplied three times. Basically, the primary source or origin is known as the root. So, all we have to do is consider "which number's cube should be taken to get the given number." The definition of cube root in Mathematics is: It is the number that must be multiplied three times to obtain the original number.