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The symbol of a square root is √ while the cube root of a number is denoted by ∛. That said, the cube root of a number y is that number whose cube results in y. Thus, we denote the cube root of y by ∛y. As an example of the cube root of 64, we denote it as 64 that gives 4³ = 4 ×4 × 4 = 64. For example:

(i) Since (4 × 4 × 4) = 64, we have ∛64 = 4, similarly

(ii) Since (6 × 6 × 6) = 216, we have ∛216 = 6

\[\sqrt[3]{64}\]

It might take you by surprise, but the process of cubing is the same as squaring, just that the number is multiplied three times instead of two. The exponent or power of the factor used for cubes is 3, which is also represented by the superscript³. As an example the cube root of 64 is 4 and it is mathematically expressed as 4³ = 4 ×4 × 4 = 64 or 7³ = 7 × 7 × 7 = 343.

The cubic function states a one-to-one function. You might be wondering how and why this is so. This is due to the reason 4 cubing negative numbers result in an outcome different from that of cubing its positive coordinate. Also, when we multiply the three negative numbers together, two of the negatives are canceled off, but one remains, so the result obtained is also negative. 9³ = 9 ×9 × 9 = 729 = (-9) × (-9) × (-9) = -729. In a similar as a perfect square, a cube number, or a perfect cube is an integer that results from cubing another integer. 729 and -729 are examples of perfect cubes.

We can easily find the cube root of a given number by using the method of factorization. To determine the cube root of a given number, proceed as given below:

Step I. write the given number as the product of primes.

Step II. Create a group in triplets of the same prime.

Step III. Identify the product of primes, selecting one from each triplet.

Step IV. This product obtained is the needed cube root of the given number.

Remember that if the grouping in triplets of the same prime factors cannot complete, then we cannot find a perfect cube root.

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If you think that finding prime or natural number factors is a tough task, then it is not. Simply, to obtain the number that you are factoring, you just need to multiply whatever number in the set of whole numbers with another within the same set. As an example:

For example, the number 3 has two factors 1 and 3. Number 8 has four factors 1, 2, 4, and 8 itself. It is very easy to factor numbers in a set of natural numbers since all numbers carry a minimum of two factors (one and itself). However, to find other factors, you will have to begin dividing the number starting from 2 and keep continuing with dividers increasing until reaching the number that was divided by 2 in the starting.

Remember that all numbers not having the remainders are factors, including the divider itself. Take an example of factorization with the number 9. Number 9 is not divisible by 2 evenly, so we will skip it. Note the numbers 4 and 5 so you know where to stop later. Number 9 is divisible by 3, so add 3 to your factors. Keep going on until you arrive at 5 (9 / 2, rounded up). Finally, you have 1, 3 and 9 as a list of factors.

Example1: Find Out the Cube Root of 216.

Solution1:

Step I. Express the given number as ∛216

Step II. Using the method of prime factorization, we have

This means: 2 × 2 × 2 × 3 × 3 × 3 = 216

Step III. Creating a group in triplets of the same prime, we have

= [2 × 2 × 2] × [3 × 3 × 3]

Thus, ∛216 = [2 × 3] = 6

Example 2: Evaluate the Cube Root of 9261.

Solution 2:

Following the same procedure of factorization, we get

This means: 3 × 3 × 3 × 7 × 7 × 7 = 9261

(3 × 3 × 3) × (7 × 7 × 7)

3 × 7 = 21

Thus, ∛9261 = [3 × 7] = 21

FAQ (Frequently Asked Questions)

1. What About the Number 64? is it a Prime Number?

No, the number 64 is not a prime number, but a composite number since the number 64 has more than 1 divisors and 64 itself. The Prime factors of number 64 are 2 which in the form of equation is expressed as: 2× 2 × 2× 2 × 2 × 2. And amazingly, even the smallest common factor, as well as the most significant common factor of 64, is number 2.

**Prime Numbers Less Than 64 Are as Follows**

1 | 5 | 7 | 59 |

11 | 13 | 17 | 53 |

19 | 23 | 29 | 47 |

31 | 37 | 41 | 43 |

**Prime Numbers Greater Than 64 Are as Follows**

67 | 71 | 73 | 79 |

83 | 89 | 97 | 101 |

103 | 107 | 109 | 113 |

131 | 127 | 137 | 139 |

2. What About the Number 4?

Number 4 is linear, which is also the 1^{st} composite number and, therefore, the 1^{st} non-prime number after the number one. The uniqueness of the number 4 is that the result of its multiplication and addition with its factor 2 gives the same result i.e. both 2 × 2 = 4 and 2 + 2 = 4 and hence 2^{2} = 4. Another peculiarity of a number 4 is its impossibility of an algebraic equation of a greater degree than four square roots using basic arithmetic and simple operations dissolve.