A perfect cube is a number that is acquired by multiplying the very same integer three times. Because that example, multiply the number 4 three times results in 64. Therefore, 64 is a perfect cube. Therefore, perfect cube = number × number × number. The cube root of 64 is 4. A number is said to be a perfect cube if it have the right to be decomposed into a product of the exact same three numbers.
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| 1. | Perfect Cube Definition |
| 2. | Cube root of Perfect Cube |
| 3. | How to find the Perfect Cube? |
| 4. | Perfect Cube Formula |
| 5. | Perfect Cube 1 come 50 List |
| 6. | Properties of Perfect Cube |
| 7. | Perfect Cube FAQs |
Perfect Cube Definition
A perfect cube is identified as the product the three very same integers. To check if a number 'N' is a perfect cube or not, we examine whether an integer, once multiplied by itself thrice, gives the number 'N'. If yes, climate it is a perfect cube. 1, 8, 27, 64 are some instances of perfect cubes. A perfect square is one that is derived by multiply a number by itself. The is different from a perfect cube. Perfect cubes deserve to be both optimistic and negative numbers. For example, -64 is a perfect cube because it is the product of multiplying -4 3 times.
Cube source of Perfect Cubes
When we say that a number is cubed, it means it is multiplied three times. Cube source is reversing the procedure of cubing a number. Because that example, once the number 5 is cubed, we gain 5 × 5 × 5, i m sorry is 125. The cube root of 125 is 5. This is because we obtain 125 on multiply the number 5 three times. The symbol for cube root is ∛. It is the exact same as the square root symbol other than that a '3' is placed in it, to display that the denotes a cube root. In the exponent form, the cube root of a number have the right to be expressed together (number)1/3.
How to find the Perfect Cube?
The perfect cube the a number deserve to be confirm by adhering to the steps offered below:
Step 1: element factorize the given number beginning from the smallest prime number (2).Step 3: Repeat the step for all the to adjust of the team of the very same three factors. If there are any factors that space left behind and do not fit right into a group of three very same factors, climate the provided number is not a perfect cube. Otherwise, the offered number is a perfect cube.Let us recognize this v an example.
There is a cheat to check if larger numbers are a perfect cube or not. To inspect that, uncover the amount of every the digits of the number repeatedly and see if that is either 0, 1, 8, or 9. If it is any one of these, then it MAY BE a perfect cube, however not have to true. To check whether that is a perfect cube we will need to again follow the same process of prime factorization method discussed above.
Note: This is a trick to check if a number might or may not it is in a perfect cube however it is not constantly guaranteed to it is in a perfect cube.
Example: The reason why this cheat is supplied is that mean we are provided a number, say, 8768, and also we space not certain if this is a perfect cube or not. So, we find the sum of the number of this number and check if that is either 0, 1, 8, or 9. If that is no either the these four numbers, then we require not factorize the number to inspect if it is a perfect cube. In other words, this trick just eliminates the process of element factorization if the amount is no equal to 0, 1, 8, or 9. In this case, the sum of the digits of the number 8768 is 29 and the amount of 2 and also 9 is 11 and also finally, 1 + 1 is same to 2. Therefore, the is no 0, 1, 8, or 9. So us can conveniently conclude the 8768 is no a perfect cube.
There is a special situation wherein we gain the sum of the digits same to either 0, 1, 8, or 9 but the number is still not a perfect cube. For example, in the number 3456, the amount of the number in 3 + 4 + 5 + 6, which is 18 and also then 1 + 8 = 9. Though the amount of the number is 9, once we factorize the number us get, 3456 = 2 × 3 × 2 × 3 × 2 × 3 × 2 × 2 × 2 × 2. So us get, 6 × 6 × 6 × 4 × 4. Since we are brief of one 4 to do a group of three exact same numbers, we deserve to say the 3456 is no a perfect cube.
The perfect cube formula is supplied to check if any number is a perfect cube or not. Intend we have actually a number, x, together that, x = y × y × y. Together per the basic theorem that arithmetic, every composite number have the right to be expressed together the product that the power of its prime factors. If the strength of every the prime factors is in the multiple of 3, climate the number is claimed to it is in a perfect cube.
The perfect cubes of number from 1 to 50 are noted in the table provided below. The perfect cubes are obtained by multiplying every number thrice by itself.
| 1 | 1 × 1 × 1 | 1 |
| 2 | 2 × 2 × 2 | 8 |
| 3 | 3 × 3 × 3 | 27 |
| 4 | 4 × 4 × 4 | 64 |
| 5 | 5 × 5 × 5 | 125 |
| 6 | 6 × 6 × 6 | 216 |
| 7 | 7 × 7 × 7 | 343 |
| 8 | 8 × 8 × 8 | 512 |
| 9 | 9 × 9 × 9 | 729 |
| 10 | 10 × 10 × 10 | 1000 |
| 11 | 11 × 11 × 11 | 1331 |
| 12 | 12 × 12 × 12 | 1728 |
| 13 | 13 × 13 × 13 | 2197 |
| 14 | 14 × 14 × 14 | 2744 |
| 15 | 15 × 15 × 15 | 3375 |
| 16 | 16 × 16 × 16 | 4096 |
| 17 | 17 × 17 × 17 | 4913 |
| 18 | 18 × 18 × 18 | 5832 |
| 19 | 19 × 19 × 19 | 6859 |
| 20 | 20 × 20 × 20 | 8000 |
| 21 | 21 × 21 × 21 | 9261 |
| 22 | 22 × 22 × 22 | 10648 |
| 23 | 23 × 23 × 23 | 12167 |
| 24 | 24 × 24 × 24 | 13824 |
| 25 | 25 × 25 × 25 | 15625 |
| 26 | 26 × 26 × 26 | 17576 |
| 27 | 27 × 27 × 27 | 19683 |
| 28 | 28 × 28 × 28 | 21952 |
| 29 | 29 × 29 × 29 | 24389 |
| 30 | 30 × 30 × 30 | 27000 |
| 31 | 31 × 31 × 31 | 29791 |
| 32 | 32 × 32 × 32 | 32768 |
| 33 | 33 × 33 × 33 | 35937 |
| 34 | 34 × 34 × 34 | 39304 |
| 35 | 35 × 35 × 35 | 42875 |
| 36 | 36 × 36 × 36 | 46656 |
| 37 | 37 × 37 × 37 | 50653 |
| 38 | 38 × 38 × 38 | 54872 |
| 39 | 39 × 39 × 39 | 59319 |
| 40 | 40 × 40 × 40 | 64000 |
| 41 | 41 × 41 × 41 | 68921 |
| 42 | 42 × 42 × 42 | 74088 |
| 43 | 43 × 43 × 43 | 79507 |
| 44 | 44 × 44 × 44 | 85184 |
| 45 | 45 × 45 × 45 | 91125 |
| 46 | 46 × 46 × 46 | 97336 |
| 47 | 47 × 47 × 47 | 103823 |
| 48 | 48 × 48 × 48 | 110592 |
| 49 | 49 × 49 × 49 | 117649 |
| 50 | 50 × 50 × 50 | 125000 |
Perfect cubes have actually some interesting properties which are detailed below.
The cube the an even number is even. Because that example, 2 is an also number, and also 23 = 8.The cube of an odd number is odd. Because that example, 3 is one odd number, and also 33 = 27.Perfect cube of a number have the right to be expressed together a amount of continually odd numbers. For example, 13 = 1 , 23 = 3 + 5. 33 = 5 + 7 + 9, 43 = 13 + 15 + 17 + 19 and also so on.The complete count of continually odd number which amount up to type a perfect cube is same to the number that is being cubed. For example, to acquire 13, just 1 strange number is involved, because that 23, it involves the sum of 2 numbers, 3 + 5. Because that 33, that is 5 + 7 + 9, where 3 odd numbers space involved. To acquire 43, we include 13 + 15 + 17 + 19. The very same pattern continues for the succeeding numbers also.Topics pertained to Perfect Cube
Check the end some exciting topics related to a perfect cube.
Example 1: use the perfect cube formula to find if 729 is a perfect cube.
Solution: The perfect cube formula is m = ∛N, whereby 'N' is the perfect cube, and 'm' is the cube source of 'N'.Given, N = 729, the cube source of 729 = ∛729.
Prime factorization of 729 = 3 × 3 × 3 × 3 × 3 × 3.Now, we club each set of 3 factors, which space the same. Therefore, we get (3 × 3) × (3 × 3) × (3 × 3).or 729 = (3 × 3)3 or 93 Therefore, cube source of 729 = 9Hence, 729 is a perfect cube.
Example 2: uncover the cube source of 512 and check if it is a perfect cube or not.
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Solution:
Let us discover the cube source of 512.Prime administrate of 512 = 8 × 8 × 8 or 83Therefore, ∛512 = 8.Since over there are specifically three factors that space the same, we have the right to say that 512 is a perfect cube.
Example 3: If the value of y3 = 1000, discover the value of 'y' using the perfect cube formula.Solution:
The perfect cube formula is m = ∛N, whereby 'N' is the perfect cube, and also 'm' is the cube source of 'N'.Given, y3 = 1000, therefore, y = ∛1000.Prime administrate of 1000 = 2 × 2 × 2 × 5 × 5 × 5or 1000 = (2 × 5)3 or 103