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Hint: Here, we will first take the cube of few of the natural numbers and then find out if the obtained cubes are equal to the natural number itself to find the required value.

Given that the number is a natural number.

We know that the natural numbers are a part of the number system, which includes all the positive integers from 1 till infinity, which is denoted by the symbol, \[\infty \].

Now we will find the cube of some of the natural numbers to find the required value.

First, we will find the cube of the natural number 1.

\[

{1^3} = 1 \times 1 \times 1 \\

= 1 \\

\]

Since we have seen from above that the cube of the natural number 1 is 1, so 1 is the required natural number whose cube is equal to itself.

We will also find the cube of some other natural number for the accuracy of our answer.

Finding the cube of the natural number 2, we get

\[

{2^3} = 2 \times 2 \times 2 \\

= 8 \\

\]

As this natural number 2 is not equal to its cube, the natural number 2 is not the required value.

Finding the cube of the natural number 3, we get

\[

{3^3} = 3 \times 3 \times 3 \\

= 27 \\

\]

Since this natural number 3 is also not equal to its cube, so the natural number 3 is also not the required value.

We will also take some larger natural number randomly and find its cube.

Let us take the number 7.

We will now find the cube the natural number 7.

\[

{7^3} = 7 \times 7 \times 7 \\

= 343 \\

\]

Again we have seen that this natural number is also not equals to its cube.

Continuing like the above, we will find out that 1 is the only natural number whose cube is equal to itself.

Therefore, the natural number whose cube is equals to itself be 1.

Note: In solving these types of questions, you should be familiar with the concept of natural numbers. Students should also take the cube of some other natural numbers carefully for more accuracy. Then use the given conditions and values given in the question, to find the required values. Also, we are supposed to write the values properly to avoid any miscalculation.

__Complete step-by-step solution:__Given that the number is a natural number.

We know that the natural numbers are a part of the number system, which includes all the positive integers from 1 till infinity, which is denoted by the symbol, \[\infty \].

Now we will find the cube of some of the natural numbers to find the required value.

First, we will find the cube of the natural number 1.

\[

{1^3} = 1 \times 1 \times 1 \\

= 1 \\

\]

Since we have seen from above that the cube of the natural number 1 is 1, so 1 is the required natural number whose cube is equal to itself.

We will also find the cube of some other natural number for the accuracy of our answer.

Finding the cube of the natural number 2, we get

\[

{2^3} = 2 \times 2 \times 2 \\

= 8 \\

\]

As this natural number 2 is not equal to its cube, the natural number 2 is not the required value.

Finding the cube of the natural number 3, we get

\[

{3^3} = 3 \times 3 \times 3 \\

= 27 \\

\]

Since this natural number 3 is also not equal to its cube, so the natural number 3 is also not the required value.

We will also take some larger natural number randomly and find its cube.

Let us take the number 7.

We will now find the cube the natural number 7.

\[

{7^3} = 7 \times 7 \times 7 \\

= 343 \\

\]

Again we have seen that this natural number is also not equals to its cube.

Continuing like the above, we will find out that 1 is the only natural number whose cube is equal to itself.

Therefore, the natural number whose cube is equals to itself be 1.

Note: In solving these types of questions, you should be familiar with the concept of natural numbers. Students should also take the cube of some other natural numbers carefully for more accuracy. Then use the given conditions and values given in the question, to find the required values. Also, we are supposed to write the values properly to avoid any miscalculation.

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