Answer:

To find the smallest number by which 16384 must be divided so that the quotient is a perfect cube, we need to determine the highest power of each prime factor in the given number.

Prime factorization of 16384:

16384 = 2^14

In order for the quotient to be a perfect cube, we need to divide 16384 by the smallest number such that the exponents of all prime factors are divisible by 3.

Exponent of 2 in 16384 is 14. To make it divisible by 3, we need to divide 14 by the highest power of 2 that is divisible by 3, which is 12 (2^12 = 4096).

Therefore, 16384 should be divided by 2^12 to obtain a perfect cube quotient.

16384 / (2^12) = 4

The quotient is 4, which is a perfect cube (2^3 = 8).

Now, let's find the cube root of the quotient.

Cube root of 4 = ∛4 = 1.5874 (rounded to four decimal places)

So, the smallest number by which 16384 must be divided to obtain a perfect cube quotient is 2^12 (4096), and the cube root of the quotient is approximately 1.5874.

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Answer:

2 is been evaluate with the numbe4