How to Find the Cube Root of 64 Using Prime Factorization
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
\[\sqrt[3]{64}\]
What is Cubing?
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.
Introduction to a Cubic Function
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.
Factorization Method to Find the Cube Root
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.
(Image to be added soon)
How to Calculate Prime Number Factors?
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.
Solved Examples of Cube Root
Let’s Quickly Find the Cube Root Using Step by Step Procedure With an Explanation
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
FAQs on Cube Root of 64 Explained with Clear Solution
1. What is the cube root of 64?
The cube root of 64 is 4 because 4 × 4 × 4 equals 64. In mathematical form, ∛64 = 4. The cube root of a number is the value that, when multiplied by itself three times, gives the original number.
2. How do you find the cube root of 64 step by step?
The cube root of 64 is 4, found by prime factorization or repeated multiplication.
- Step 1: Prime factorize 64 = 2 × 2 × 2 × 2 × 2 × 2
- Step 2: Group into triples: (2 × 2 × 2)(2 × 2 × 2)
- Step 3: Each triple gives one 2, so 2 × 2 = 4
3. Why is the cube root of 64 equal to 4?
The cube root of 64 equals 4 because 4 cubed equals 64. Since 4³ = 4 × 4 × 4 = 64, 4 is the number that produces 64 when multiplied by itself three times.
4. What is the cube root formula?
The cube root formula is ∛x = x^(1/3). This means the cube root of any number x can be written using exponents as x raised to the power of 1/3. For example, 64^(1/3) = 4.
5. Is the cube root of 64 a rational number?
Yes, the cube root of 64 is a rational number because it equals 4. Since 4 is an integer and can be written as 4/1, ∛64 is rational.
6. What is the difference between the square root and cube root of 64?
The square root of 64 is 8, while the cube root of 64 is 4. The square root finds a number multiplied twice (8 × 8 = 64), whereas the cube root finds a number multiplied three times (4 × 4 × 4 = 64).
7. How do you write the cube root of 64 in exponential form?
The cube root of 64 in exponential form is 64^(1/3). Using exponent rules, 64^(1/3) simplifies to 4, since 4³ = 64.
8. What are the factors of 64 used to find its cube root?
The prime factors of 64 are 2 × 2 × 2 × 2 × 2 × 2. When grouped into triples (2 × 2 × 2)(2 × 2 × 2), each group gives one 2, and multiplying them gives 4, the cube root of 64.
9. Is 64 a perfect cube?
Yes, 64 is a perfect cube because it can be written as 4³. Since 4 × 4 × 4 = 64, 64 has an integer cube root, which is 4.
10. Can you give another example similar to the cube root of 64?
Yes, a similar example is the cube root of 27, which equals 3 because 3³ = 27. Like 64, the number 27 is also a perfect cube since its cube root is a whole number.
