The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which Class 8

The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which Class 8

Question 1.

The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.

Solution:

4104 = 2³ + 16³

and 4104 = 9³ + 15³

13832 = 2³ + 24³

and 13832 = 18³ + 20³

Question 2.

Can you tell what this sum is without doing the calculation? Class 8

Solution:

Yes, it is equal to 10³ = 1000.

Question 3.

Find the cube roots of these numbers : Class 8

(i) [latex]\sqrt[3]{64}[/latex]

(ii) [latex]\sqrt[3]{512}[/latex]

(iii) [latex]\sqrt[3]{729}[/latex]

Solution:

(i) 64 = 4 × 4 × 4. So, [latex]\sqrt[3]{64}[/latex] = 4.

(ii) 512 = 8 × 8 × 8. So, [latex]\sqrt[3]{512}[/latex] = 8.

(iii) 729 = 9 × 9 × 9. So, [latex]\sqrt[3]{729}[/latex] = 9.