The smallest number that can be expressed as the sum of two cubes of natural numbers is 1729. To understand why this is the case, let's break down the problem step by step:

Step 1: Understanding Cubes
Cubing a number means multiplying it by itself twice. For example, the cube of 2 is 2^3 = 2 × 2 × 2 = 8.

Step 2: Exploring Cube Sums
We are looking for a number that can be expressed as the sum of two cubes. Let's consider the cube of natural numbers up to a certain limit and find their sums:

- Cube of 1: 1^3 = 1
- Cube of 2: 2^3 = 8
- Cube of 3: 3^3 = 27
- Cube of 4: 4^3 = 64
- Cube of 5: 5^3 = 125

Now, let's consider the sums of these cubes:
- Sum of cubes of 1 and 2: 1 + 8 = 9
- Sum of cubes of 1 and 3: 1 + 27 = 28
- Sum of cubes of 1 and 4: 1 + 64 = 65
- Sum of cubes of 1 and 5: 1 + 125 = 126
- Sum of cubes of 2 and 3: 8 + 27 = 35
- Sum of cubes of 2 and 4: 8 + 64 = 72
- Sum of cubes of 2 and 5: 8 + 125 = 133
- Sum of cubes of 3 and 4: 27 + 64 = 91
- Sum of cubes of 3 and 5: 27 + 125 = 152
- Sum of cubes of 4 and 5: 64 + 125 = 189

As we can see, none of these sums give us the smallest number that can be expressed as the sum of two cubes of natural numbers.

Step 3: Discovering the Smallest Number
To find the smallest number, we need to continue exploring the sums of cubes. Eventually, we find that the sum of cubes of 9 and 10 gives us the desired number:
- Sum of cubes of 9 and 10: 9^3 + 10^3 = 729 + 1000 = 1729

Therefore, 1729 is the smallest number that can be expressed as the sum of two cubes of natural numbers.

Conclusion:
The smallest number that can be expressed as the sum of two cubes of natural numbers is 1729. This number is obtained by calculating the sum of cubes of 9 and 10.

Because 1729 is expressed as cube of two number that is 10 and 9. cube of 10= 1000 , cube of 9= 729 and the sum is 1729.but others are not expressed in this way