Cubes and Cube Roots

Introduction 

The first thing we think about when we hear the word " Cube " is the image given below.

But the word "Cube" also refers to a number.

Cube of a number is obtained by multiplying a number twice with itself.

So, the cube of 5 would be 5 x 5 x 5 = 125.

An interesting story about the cube of a number 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 an interesting conversation, Hardy said that the cab he took had the most boring number plate. When Ramanujan asked him the number, he replied 1729. In a flash, Ramanujan said that this was not an uninteresting number at all, but a most interesting number. When Hardy asked him why Ramanujan 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. The 2 sets are given below:

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

So, the two sets of numbers are ( 9, 10 ) and ( 1, 12 ). Let us learn more about cube and cube roots.

The Big idea: Cubes and cube roots

Definition of the Cube of a number

The cube of a number is the equivalent of raising it to the exponent 3.

Example:

When you need to find the cube of 6, you would write it as:

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

Now suppose, the cube of a random number is given. From this cube we need to find that number whose cube is given. This is where cube roots come in.

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

Before we go into this method, let us revise a few definition that will be needed: 

  • A prime number is a number which is divisible only by 1 and by itself.
  • So, 2, 5, 7, 11 are all prime numbers.
  • The process of factorisation is the method in which a number is broken down to a product of prime numbers. These numbers 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?

The following method can be used to find the cube root without much calculation.

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.

Activity

Activity Name Sub_CubeRoots_Activity
Item Name 1 Sub_CubeRoots_Item1
Item Name 2 Sub_CubeRoots_Item2
Item Name 3 Sub_CubeRoots_Item3
Item Name 4 Sub_CubeRoots_Item4
Item Name 5 Sub_CubeRoots_Item5

Frequently Asked Questions

1. How do you find the cubed root of a cube?

Ans. The following example shows hoe to calculate the cube root. We are finding the cube root of 438976. 

  • 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.

2. What are cubed root in math?

Ans. 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:

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

3. Do cube roots have Plus or minus?

Ans. Unlike a square root, the cube root of a number can be negative. So a cube root of a number can be any real number, be it positive, negative or zero. 

  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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