Find the cube root of: - 27 125 .

The given rational number is - 27 125 . We need to find the cube root of the given number. If m is any positive integer, then - m is a negative integer. Since, - m 3 = - m × - m × - m = - m 3 ∴ - m 3 3 = m 3 - 3 = - m Cube - root of - m 3 = - Cube - root of m 3 Prime factorisation of 27 125 is 3 27 3 9 3 3 1 and 5 125 5 25 5 5 1 Now, - 27 125 = - 3 × 3 × 3 5 × 5 × 5 Take one prime number from each triplet and multiply all the prime numbers obtained to get the required cube root. That is, - 27 125 3 = - 3 5 . Hence, the cube root of given number is - 3 5 .

Find the cube root of: - 64 343 .

The given rational number is - 64 343 . We need to find the cube root of the given number. If m is any positive integer, then - m is a negative integer. Since, - m 3 = - m × - m × - m = - m 3 ∴ - m 3 3 = m 3 - 3 = - m Cube - root of - m 3 = - Cube - root of m 3 Prime factorisation of - 64 343 is 2 64 2 32 2 16 2 8 2 4 2 2 1 and 7 343 7 49 7 7 1 Now, - 64 343 = - 2 × 2 × 2 × 2 × 2 × 2 7 × 7 × 7 Take one prime number from each triplet and multiply all the prime numbers obtained to get the required cube root. That is, - 64 343 3 = - 2 × 2 7 = - 4 7 . Hence, the cube root of given number is - 4 7 .

Find the cube root of: - 512 343 .

The given rational number is - 512 343 . We need to find the cube root of the given number. If m is any positive integer, then - m is a negative integer. Since, - m 3 = - m × - m × - m = - m 3 ∴ - m 3 3 = m 3 - 3 = - m Cube - root of - m 3 = - Cube - root of m 3 Prime factorisation of - 512 343 is 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1 and 7 343 7 49 7 7 1 Now, - 512 343 = - 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 7 × 7 × 7 Take one prime number from each triplet and multiply all the prime numbers obtained to get the required cube root. That is, - 512 343 3 = - 2 × 2 × 2 7 = - 8 7 . Hence, the cube root of given number is - 8 7 .