If you multiply a number by itself and then by itself again, the result is a cube number.

**Cubes and
Cube Roots**

If you multiply
a number by itself and then by itself again, the result is a cube number.

This means
that a cube number is a number that is the product of three identical numbers.

If ** n** is a number, its cube is represented by

Cube numbers
can be represented visually as 3D cubes comprising of single unit cubes. Cube numbers
are also called as perfect **cubes**. The perfect cubes of natural numbers are
1, 8, 27, 64, 125, 216, ... and so on.

*Ramanujan Number*** - ****1729 = 12 ^{3}+1^{3} = 10^{3}+9^{3}**

Once Professor
Hardy went to see Ramanujan when he was ill at Putney, riding in taxi cab number
1729 and said that the number seemed a dull one, and hoped it was not an unfavourable
omen. “No,” replied Ramanujan and he completed saying “It is a very interesting
number. Infact, it is the smallest number expressible as the sum of two cubes in
two different ways.” 4104, 13832, 20683 are a few more examples of Ramanujan-Hardy
numbers.

** **

__1. Properties
of cubes of numbers__

**Note**

A perfect cube does not end with two zeroes.

The cube of a two digit number may have 4 or 5 or 6 digits in it.

**Try these**

Find the ones
digit in the cubes of each of the following numbers.

1. 12

2. 27

3. 38

4. 53

5. 71

6. 84

**Solution:**

**(i) 12**

12 ends with 2, so its cube ends with 8 i.e, ones digit in 12^{3}
is 8.

**(2) 27**

27 ends with 7, so its cube end with 3. i.e., ones digit in 27^{3}
is 3.

**(3) 38**

38 ends with 8, so its cube ends with 2 i.e, ones digit in 38^{3}
is 2.

**(4) 53**

53 ends with 3, so its cube ends with 7. i.e, ones digit in 53^{3}
is 7.

**(5) 71**

71 ends with 1, so its cube ends with 1. i.e, ones digit in 71^{3}
is 1

**(6) 84**

84 ends with 4, so its cube ends with 4. i.e, ones digit in 84^{3}
is 4.

** **

__2. Cube
root__

The cube
root of a number is the value that when cubed gives the original number.

For example,
the cube root of 27 is 3 because when 3 is cubed we get 27.

**Notation:**

The cube
root of a number *x* is denoted as

^{3}√*x*
(or)* x*^{1/3}

Here are
some more cubes and cube roots:

^{3}√1 = 1 since 1^{3} = 1, ^{3}√8
= 2 since 2^{3} = 8,

^{3}√27 = 3 since 3^{3} = 27, ^{3}√64 = 4 since 4^{3} = 64,

^{3}√125 = 5 since 5^{3} = 125 and so on.

__Example 1.32__

Is 400 a
perfect cube?

**Solution:**

By prime
factorisation, we have 400 = __2 × 2 × 2__ × 2 × 5 × 5

There is
only one triplet. To make further triplets, we will need two more 2’s and one more
5.

Therefore,
400 is not a perfect cube.

__Example 1.33__

Find the
smallest number by which 675 must be multiplied to obtain a perfect cube.

**Solution:**

We find that,
675 = 3 × 3 × 3 × 5 × 5 …………….(1)

Grouping
the prime factors of 675 as triplets, we are left over with 5 × 5.

We need one
more 5 to make it a perfect cube.

To make 675
a perfect cube, multiply both sides of (1) by 5.

675×5=3×3×3×5×5×5

3375=3×3×3×5×5×5

Now, 3375
is a perfect cube. Thus, the smallest required number to multiply 675 such that
the new number perfect cube is 5.

**Think**

In this question, if the word ‘multiplied’ is replaced by the word
‘divided’, how will the solution vary?

** **

__3. Cube
root of a given number by Prime Factorisation__

Step 1: Resolve
the given number into the product of prime factors.

Step 2: Make
triplet groups of same primes.

Step 3: Choosing
one from each triplet, find the product of primes to get the cube root.

__Example 1.34__

Find the
cube root of 27000.

**Solution:**

By prime
factorisation, we have 27000 = 2× 2 ×2× 3 × 3 × 3 × 5 × 5 ×5

∴ ^{3}√27000 =
2×3×5
=30

__Example 1.35__

Tags : Numbers | Chapter 1 | 8th Maths , 8th Maths : Chapter 1 : Numbers

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8th Maths : Chapter 1 : Numbers : Cubes and Cube Roots | Numbers | Chapter 1 | 8th Maths

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