Question

# Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?

Solution

## (i) We have:     $\because$ The next number to be subtracted is 91, which is greater than 5. $\therefore$ 130 is not a perfect cube. However, if we subtract 5 from 130, we will get 0 on performing successive subtraction and the number will become a perfect cube. If we subtract 5 from 130, we get 125. Now, find the cube root using successive subtraction. We have:     $\because$ The subtraction is performed 5 times. $\therefore$ $\sqrt[3]{125}=5$  Thus, it is a perfect cube. (ii) We have:     $\because$ The next number to be subtracted is 161, which is greater than 2. $\therefore$ 345 is not a perfect cube. However, if we subtract 2 from 345, we will get 0 on performing successive subtraction and the number will become a perfect cube. If we subtract 2 from 345, we get 343. Now, find the cube root using successive subtraction. $\because$ The subtraction is performed 7 times. $\therefore$ $\sqrt[3]{343}=7$  Thus, it is a perfect cube. (iii) We have:     $\because$ The next number to be subtracted is 271, which is greater than 63. $\therefore$ 792 is not a perfect cube. However, if we subtract 63 from 792, we will get 0 on performing successive subtraction and the number will become a perfect cube. If we subtract 63 from 792, we get 729. Now, find the cube root using the successive subtraction. We have:     $\because$ The subtraction is performed 9 times. $\therefore$ $\sqrt[3]{729}=9$  Thus, it is perfect cube.MathematicsRD Sharma (2019, 2020)All

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