Question

which of the following number is a perfect square as well as a perfect cube? 64, 1000, 8, 125​

Answer 1

Answer:

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Answer 2

Given,

The numbers are = 64,1000,8,125

To find,

The number which is a perfect square as well as a perfect cube.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

First number :

Square root = √64 = 8

Cube root = ³√64 = 4

Second number :

Square root = √1000 = 31.622

Cube root = ³√1000 = 10

Third number :

Square root = √8 = 2.828

Cube root = ³√8 = 2

Fourth number :

Square root = √125 = 11.180

Cube root = ³√125 = 5

So, from the above mentioned calculations, it is clear that only the of square root and cube root of 64 produces a whole number in both cases. So, 64 is the number which is perfect square and perfect cube.

Also, this can be derived from doing the prime factorization of each number. If the each unique prime factor have an exponent which is divisible by 2 then the main number will be perfect square and if the each unique prime factor have an exponent which is divisible by 3 then the main number will be perfect cube.

Example :

64 = 2×2×2×2×2×2 = 2⁶

There is only one unique prime factor in the above mentioned prime factorization, which is 2 and it's exponent is 6.

This 6 is divisible by both 2 and 3. That's why, the main number or 64 is both a perfect square and a perfect cube.

Hence, 64 is a perfect square and a perfect cube.