# Cubes and cube roots

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Introduction: We know of those three-dimensional rectangular prisms that have all three sides equal. They are called cubes. But the word cube also refers to a number obtained by multiplying any number by itself twice over. So, the cube of 5 would be 5 x 5 x 5 = 125. The most famous anecdote about a cube is the one concerning the great Indian mathematician Ramanujan and his friend Hardy.

Ramanujan was unwell, and his friend Hardy came to visit him. To cheer up his friend with some desultory conversation, Hardy made his opening remark that the cab he took to reach had a most boring number plate. When Ramanujan asked him the number, he replied 1729. In a flash, the great mind of Ramanujan prompted him to say that this was not an uninteresting number at all, but a most interesting number. When Hardy asked him why Ramanujan replied that 1729 replied that 1729 was the smallest number that could be expressed as the sum of two unique cubes in two ways!! Since then, 1729 has been known as the Hardy – Ramanujan number. Now what are the two sets of two unique numbers, I hear you ask. Well here they are:

\begin{align} 1729={12^3} + {1^3}\\ 1729 = {10^3} + {9^3} \end{align}

So, the two sets of numbers are 9, 10 and 1, 12!! Apart from this interesting story, there are lots of facet to cubes and cube roots and let us understand a few of them.

## The Big idea: Cubes and cube roots

### A simple idea: The Power of 3

The cube of a number is the equivalent of raising it to the exponent 3. So, when you need to find the cube of 6, you would write it as:

\begin{align}{6^3} = 6 \times 6 \times 6 = 216\end{align}

The inverse function of the cube is the cube root. So, if we examine the number 8, it is the cube of 2, and the cube root of 8 would be written in the following manner:

\begin{align}{8^{\left( {\frac{1}{3}} \right)}} = 2\end{align}

Once again, the number line comes to the rescue as an invaluable tool to try and visualize how a number grows when it is multiplied by itself thrice. Here, have a look; ### An Interesting Fact about Cubes

Let us examine the first five numbers 1 to 5. Their cubes can be written thus:

\begin{align} &{1^3} = 1 \times 1 \times 1 = 1,{\text{ which is an odd number}}\\ &{2^3} = 2 \times 2 \times 2 = 8,{\text{ which is an even number}}\\ &{3^3} = 3 \times 3 \times 3 = 27,{\text{ which is an odd number}}\\ &{4^3} = 4 \times 4 \times 4 = 64,{\text{ which is an even number}}\\ &{5^3} = 5 \times 5 \times 5 = 125,{\text{ which is an odd number}} \end{align}

So, the thing to be noted here is that the cubes of odd numbers are odd, while the cubes of even numbers are even.

### The prime factor method for finding cube roots

A prime number is a number which is divisible only by 1 and by itself. So, 2, 5, 7, 11 are all prime numbers. And we also know about the process of factorisation in which a number is broken up into several numbers which when multiplied together give that number. Let us see the example of how to factorise 42

\begin{align} &42 = 2 \times 21\\ &\;\;\;\; = 2 \times \left( {3 \times 7} \right)\\ &\;\;\;\; = 2 \times 3 \times 7 \end{align}

All of the above numbers 2, 3 and seven are referred to as factors of 42, and incidentally, they are all prime factors as well, which means all the factors are prime numbers.

When a cube is broken into its prime factors, it helps to find the cube root as well. Let us take the example of 64, which is the cube of 4.

\begin{align} & 64 = 2 \times \left( {32} \right) \hfill \\ & \;\;\;\; = 2 \times 2 \times \left( {16} \right) \hfill \\ & \;\;\;\; = 2 \times 2 \times 2 \times \left( 8 \right) \hfill \\ & \;\;\;\; = 2 \times 2 \times 2 \times 2 \times \left( 4 \right) \hfill \\ & \;\;\;\; = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \hfill \\ & \;\;\;\; = \left( {2 \times 2 \times 2} \right) \times \left( {2 \times 2 \times 2} \right) \hfill \\ & \;\;\;\; = {2^3} \times {2^3} \hfill \\ & \;\;\;\; = {\left( {2 \times 2} \right)^3} \hfill \\ & \;\;\;\; = {\left( 4 \right)^3} \hfill \\ \end{align}

Therefore, by breaking up 64 into its prime factors, we can know that it is the cube of 4, so its cube root would be 4.

## Why is it important?

### A Simple Application of Cubes

Let us say you have a huge box of side 1 meter. You need to pack it with smaller cubical boxes of side 10 cm each. How many of these boxes do you think you can pack into the bigger box?

\begin{align} & {\text{The volume of the biggest box}} = 1{\text{ m}} \times 1{\text{ m}} \times 1{\text{ m}} \hfill \\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad = 100{\text{ cm}} \times 100{\text{ cm}} \times 100{\text{ cm}} \hfill \\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad = 10,00,000{\text{ cm}}{^3} \hfill \\ \end{align}

\begin{align} & {\text{The volume of each small box}} = 10{\text{ cm}} \times 10{\text{ cm}} \times 10{\text{ cm}} \hfill \\ & \qquad \qquad \qquad \qquad \qquad \qquad \;\;\, = 1000{\text{ cm}}{^3} \hfill \\ \end{align}

\begin{align}{\text{So, the number of boxes}} = \frac{{1000000}}{{1000}} = 1000\end{align}

You can pack 1000 small boxes into the bigger box.

## Tips and Tricks

• Here are the cubes of numbers from 0 to 9:

03   =   0
13   =   1
23   =   8
33   =   27
43   =   64
53   =   125
63   =   216
73   =   343
83   =   512
93   =   729

Interestingly, if you have any other 2-digit or greater number, looking at its units digit, you can tell with certainty what the units digit will be for that number’s cube or cube root
For instance, any number ending with 4 will have 4 in the units place of its cube too.
Any number ending in 7 will always have 3 in the units place of its cube.
Conversely, a number ending in 3 will have 7 in the units place of its cube root.

• We can use this to estimate the cube root of a large number
What’s the cube root of 4,38,976?

Since this is a 6-digit number, the cube root will be two digits.
Since the number ends in 6 it’s cube root will have 6 in the units place.
Ignore the last three digits to estimate the tens digit of the cube root. We are left with 438.
The cube root of 438 will be between 7 and 8 (referring to the cubes list given above).
So, for the number we are given, the cube root will be between 70 and 80.
So, the cube root is 76.