Perfect Cube


 
 
Concept Explanation
 

Perfect Cube

Cube of a number:  When we multiply a number three times by itself, then the number is said to be cubed, and the product is called the cube of that number or the number raised to the power of 3.

In general, the cube of a number xtimes xtimes x=x^{3}

For example,

Cube of 1 = 1times 1times 1=1^{3}

Cube of 2 = 2times 2times 2=2^{3}

Cube of 3 = 3times 3times 3=3^{3}

and so on.

Using the general formula, the cubes of the first 15 natural numbers are written in the following table

Number Cube Number Cube Number Cube
1 1 6 216 11 1331
2 8 7 343 12 1728
3 27 8 512 13 2197
4 64 9 729 14 2744
5 125 10 1000 15 3375

From this table, it is clear that 1,8,27,64,...,3375 are the cubes of some natural numbers. Such numbers are called perfect cubes.

Perfect Cube: A natural number n is a perfect cube if there exist a natural number m such that:

                                n=mtimes mtimes m

                       Or     n=m^{3}

Hence, a natural number that can be expressed as a product of triplets of equal factors is known as a perfect cube.

Cube Root: As we know, the square root is the opposite of a square. In the same way, the cube root is the opposite of a cube. For example,

                        Cube of 8 = 8^{3}=8times8times8 = 512

                       Cube root of 512 = 8

                       Cube of 7 = 7^{3}=7times7times7 = 343

                       Cube root of 343 = 7

In general, the cube of x=x^{3}=xtimes xtimes x  and the cube root of x^{3}  is x. For any number x, the cube root is denoted by sqrt[3]{x}.

Hence sqrt[3]{512}=8  and   sqrt[3]{343}=7

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Sample Questions
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Question : 1

If 72 K is a perfect cube, then the value of K is :

Right Option : C
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Question : 2

The length of each edge of a cube is 9 m. The volume of the cube, in m^3, is :

Right Option : D
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Explanation
Question : 3

sqrt [3]{216} is :

Right Option : C
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