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Cube is defined as the multiplication of a number to itself three times and Cube Root is defined as the reverse of Cube. Or we can say that the cube root of a number provides us with a number which when cubed will provide the same number and is called a perfect cube. There are different methods of finding out the cube and cube root.
| Table of Content |
Keyterms: Cube, Cube root, perfect cube, three-dimensional geometry, Length, Breadth, Height, ice cubes, dice, cubical boxes
Read More: Difference between Cube and Cuboid
Cube and Cube Root
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In three-dimensional geometry, a Cube is a geometrical shape which has three dimensions such as Length, Breadth and Height. All these three dimensions are equal in length in case of cube and hence are denoted as Sides (a) or edges of the cube. This cube is in square shape when we consider it as a Two-dimensional figure.
Cube
Objects like ice cubes, dice, cubical boxes, and certain rooms are also designed in perfect cubical patterns.
Read More: Cubes and Cube roots
Symbol for Cube Root
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The symbol of cube root is given by‘3√’ and (given number)3 is the symbol for finding cube of that given number.
Example:
- 3√27 = 3, this means that if 3 is multiplied thrice to itself or cubed, it will become 27.
- If we write (5)3, then it means that we need to cube or multiply 5 thrice to itself: 5×5×5= 125.
Symbol for Cube Root
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| Circles | Triangles | Similarities of Triangle |
| Pythagoras Theorem | Trigonometric Identities | Construction Formula |
Cube Root Formula
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Cube root of a given number with the same or different digits is calculated by the method of Prime Factorization where we factor the given number in the smallest prime factors and then cube the same numbers together and multiply them to get the cube root.
Example:
(i) To find cube root of 512:
512= 2×2×2×2×2×2×2×2×2
→ 2×2×2 = 8.
(ii) To find cube root of 216:
216= 2×2×2×3×3×3
→ 2×3= 6.
Also Read: Section Formula
Cubes and Perfect Cubes Roots
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Every number cannot have its perfect cube but every number has a cube number. Cube is calculated by simply multiplying the number thrice to itself and cube root is calculated by prime factorization of the number. Some of the cubes and perfect cubes roots are tabulated below:
| Number | Cube |
|---|---|
| 1 | 1×1×1= 1 |
| 2 | 2×2×2=8 |
| 3 | 3×3×3= 9 |
| 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 |
Some of the Perfect cubes and its cube root are tabulated below.
| Number | Cube root |
|---|---|
| 1 | 3√1= 1 |
| 8 | 3√8= 2 |
| 9 | 3√27= 3 |
| 64 | 3√64= 4 |
| 125 | 3√125 =5 |
| 216 | 3√216 =6 |
| 343 | 3√343= 7 |
| 512 | 3√512= 8 |
| 729 | 3√729= 9 |
| 1000 | 3√1000= 10 |
| 1331 | 3√ 1331= 11 |
| 1728 | 3√1728= 12 |
| 2197 | 3√2197= 13 |
| 2744 | 3√2744= 14 |
| 3375 | 3√3375= 15 |
Read More: Roots of Quadratic Equations
Things to Remember
- Cube is defined as the multiplication of a number to itself three times.
- Cube Root is defined as the reverse of Cube.
- A perfect cube is where when a cube root is taken for that number, it results in a natural number.
- Cube root of a given number is calculated by the method of Prime Factorization.
- Every number cannot have its perfect cube but every number has a cube number.
Also Read:
Sample Questions
Ques: Is 18432 a perfect cube? If not, tell the smallest which can be multiplied to make it a perfect cube. [2 marks]
Ans: If we factorize 18432, we will get :
18432 = 8×8×8×6×6
So, only 8 can be grouped into a single group but not 6. Hence it is not a perfect cube and to make it a perfect cube one 6 must be multiplied to the given number. Then it will be:
8×8×8×6×6×6 = 8 ×6= 48 = 3√110,592
Ques: Is there any other method to find the cube root of a number carrying different digits? [2 marks]
Ans: Yes, there are different methods of calculating the cube root of a given number like through Prime Factorization, by Long division method and by estimating the number. Long division method is studied in higher classes.
Ques: What is the cube root of 3375? [2 marks]
Ans: To find the cube root of 3375, we need to do the prime factorization:
3375 = 3×3×3×5×5×5
→ 3×5 = 15
Ques: Find the cube root of 27000? [2 marks]
Ans: To find the cube root of 27000, we need to do the prime factorization:
27000 = 3×3×3×5×5×5×2×2×2
→ 3×5×2= 30
Ques: What is the difference between cube root and square root of a number? [2 marks]
Ans: Cube root is the reserve of a cube where the given number has to factor and group the number into three. Square root is the reserve of squares where the given number has to factor and group the number into twice.
Ques: How to calculate the cube of 4.5? [2 marks]
Ans: The cube of 4.5 can be calculated by multiplying the number thrice by itself.
4.5×4.5×4.5= 91.125
Ques: What is the cube root and square root of 64? [2 marks]
Ans: Cube root of 64:
64= 2×2×2×2×2×2
→ 2×2= 4
Square root of 64:
64= 2×2×2×2×2×2
→ 2×2×2 = 8
Ques: If one side of a cube is 6cm. What is the volume of that cube? [2 marks]
Ans: Side of a cube = 6cm.
Therefore volume of cube= 6×6×6= 216cm³.
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