What is the smallest positive integer K such that 168*k is the cube

When we are asked to determine a number to make an expression be the square, cube, etc of a positive integer, we always have to factorize that expression and then find the number that make the exponents of all factors equal 2 (square), 3 (cube), etc.

Let's go ahead. I will apply the mentioned steps to find the answer:

1) Factorize the expression:

From the given expression, only the number 168 can be factorized. The new expression would equal (2^3)*(3)*(7)*(k)^1/2

2) Find the number that makes the exponents of all factors equal 3 :

for the factor 2, we already have the cube of it in the expression, but for 3 and 7, we need additionally 3^2 and 7^2.

But since we are given the root of k, we would need the number to have the factors 3^4 and 7^4 so that in overall we can have (2^3)*(3^3)*(7^3)

So the right answer would be 3^4*7^4=194481

Since we don't have this number among the options, we can conclude that the problem hasn't been formulated correctly.

But going a bit further, if we calculate 3^2 and 7^2, then we get 441, which is option E

Therefore, we can assume that option E is the correct answer if expression had been 168*K and not 168*(K)^1/2

fauji: Please modify the problem expression accordingly!