CLASS-7
CUBE ROOT

CUBE ROOT -

A cube root is a special value that, when we multiply three times, gives us the desired number. Thus, a perfect cube is a cube of a whole number.

A cube root of a number x is a value that, when multiplied by itself three times, gives x. It is denoted as βˆ›x​.

A cube root is a special value that, when we multiply three times, gives us the desired number. Thus, a perfect cube is a cube of a whole number. Cube root is the inverse process of calculating the cube of a number. It is denoted by the symbol β€˜βˆ›β€™. Let us see some examples here now.

To find the cube root of a nunumber 7, we want a number which when multiplied thrice with itself shall give 27. We can write,

                 27 = 3 Γ— 3 Γ— 3 = 3Β³

Taking cubic root on both the sides; or βˆ›27 = βˆ›3Β³

Therefore, the cube-root of 27 is 3.

Please note that we will only consider the positive values cube roots of the natural numbers.

For Example:-

  • βˆ›27 = 3 because 3 Γ— 3 Γ—3 = 3Β³= 27.
  • βˆ›8 = 2 because 2 Γ— 2 Γ— 2 = 2Β³= 8.
  • βˆ›125 = 5 because 5 Γ— 5 Γ— 5 = 5Β³= 125.
  • βˆ›βˆ’64 = βˆ’4 because βˆ’4 Γ— βˆ’4 Γ— βˆ’4 = (-4)Β³= βˆ’64.
  • βˆ›2 β‰ˆ 1.2599 (since 2 is not a perfect cube, its cube root is an irrational number).
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Some Other Way Of Understanding Cube Root:-

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 1. Volume Perspective (Geometric Interpretation):-

Imagine you have a cube with a total volume of x. The cube root of x tells you the length of each side of the cube.

  • Example:- If the volume is 27 cubic units, then the side length is 3 units because 3 Γ— 3 Γ— 3 = 27.

 2. Repeated Multiplication:-

The cube root of a number is the value that you multiply by itself three times to get the original number.

  • Example:-   βˆ›64 = 4 because 4 Γ— 4 Γ— 4 = 4Β³ = 64 and βˆ›64 = βˆ›4Β³= 4

 3. Inverse of Cubing:-

Just like squaring and square roots are inverse operations, cubing and cube roots undo each other:

  • 2Β³ = 8 β†’ βˆ›8 = βˆ›2Β³= 2
  • 5Β³ = 125 β†’ βˆ›125 ​= βˆ›5Β³ = 5

  4. ο»ΏGraphical Understandingο»Ώ:-

If you graph y = xΒ³, the cube root is just moving backward along that curve. Unlike square roots, cube roots can be negative:

  • βˆ›βˆ’8 = βˆ’2 because (βˆ’2) Γ— (βˆ’2) Γ— (βˆ’2) = βˆ’8.

  5. Real-Life Example:-

Think about a sugar cube or a Rubik’s Cube:-  If you know the total number of smaller cubes inside, the cube root tells you how many cubes line up along each edge.

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