Question

What is the smallest positive integer, other than $1$, that is both a perfect cube and a perfect fourth power?

Answer 1

There are two approaches to translate this inquiry, to be specific:
    You need to know a number which can go about as the ideal square root and also the ideal block root.
    You need to know a number which is an ideal square and in addition an ideal 3D shape of a whole number.
    In the primary case, the arrangement is straightforward. Any non-negative whole number is an ideal square root and in addition a flawless solid shape foundation of a bigger number.
    A non-negative whole number, say 0, is the ideal square foundation of 0 and additionally an immaculate shape base of 0. This remains constant for all non-negative numbers starting from 0 i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
    In the second case as well, the arrangement is straightforward however it involves a more legitimate approach than the primary choice.
    A flawless square is a number which contains prime variables having powers which are a different of 2. So also, a flawless block is a number which includes prime variables having powers which are a numerous of 3. Any number which includes prime components having powers which are a various of 6 will be the answer for your inquiry; a case of which would be 64 which is the ideal square of 8 and an ideal 3D shape of 4. For this situation, the number 64 can be spoken to as prime variables (i.e. 2^6) having powers (i.e. 6) which are a different of 6.

Answer 2

First you need to find the least common multiply of the powers; in this case 3 and four.
lcm=3x4=12
Now, the lcm12 is the number you need to raise the next smallest positive integer so the result will a be perfect cube and a perfect fourth power.
And we also know that the next smallest positive integer after 1 is 2; therefore the number is:
$2^{12} =4096$
$\sqrt[3]{4096} =16$ and $\sqrt[4]{4096} =8$