To determine whether 9000 is a perfect cube, we need to find the prime factorization of 9000.

9000=23×32×539000=23×32×53

In a perfect cube, each prime factor must have an exponent that is a multiple of 3. Here, the exponent of 2 is 3, the exponent of 3 is 2 (not a multiple of 3), and the exponent of 5 is 3.

Since the exponent of 3 is not a multiple of 3, 9000 is not a perfect cube.

To find the smallest number by which  should be divided so that the quotient is a perfect cube, we need to factorize  into its prime factors.

42592=25×=25×133

To make the quotient a perfect cube, we need to divide by 2^2 (since the smallest multiple of 3 greater than or equal to 5 is 3).

42592÷22=42592÷4=÷22=42592÷4=10648

Now, let's check if 10648 is a perfect cube:

10648=23×=23×133

So, the smallest number by which should be divided to get a perfect cube quotient is 4, and the quotient is

The cube root of  is 36 because 36×36×36=×36×36=

So,  is indeed a perfect cube, as it can be expressed as the cube of a whole number (36).

To find the smallest number by which 704 should be divided to obtain a perfect cube, we need to factorize 704 and find the prime factors.

Prime factorization of 704: 704=26×11704=26×11

Now, to obtain a perfect cube, we want to form triplets of the prime factors. In this case, we can group the factors as follows: 704=(22×11)×(22×11)×22704=(22×11)×(22×11)×22

Now, we need to find the product of the remaining factors, which is 2222.

So, the smallest number by which 704 should be divided to obtain a perfect cube is 22=422=4.