Introduction:
We are given the equation x^4 (1-2k)x^2 (k^2-1) =0 and we need to find the values of k for which the equation has no real roots.

Analysis:
To solve this problem, we can use the concept of discriminant. The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. If the discriminant is negative, it means that the quadratic equation has no real roots.

Solution:
We can break down the given equation into two parts:
1) x^4 (1-2k) = 0
2) x^2 (k^2-1) = 0

Case 1: x^4 (1-2k) = 0
For this equation to have no real roots, the discriminant of x^4 (1-2k) must be negative. Let's calculate the discriminant:

D = 0^2 - 4(1-2k)(0)
D = -4(1-2k)(0)
D = 0

Since the discriminant is zero, it means that this equation will always have at least one real root. Therefore, there are no values of k for which this equation has no real roots.

Case 2: x^2 (k^2-1) = 0
For this equation to have no real roots, the discriminant of x^2 (k^2-1) must be negative. Let's calculate the discriminant:

D = 0^2 - 4(k^2-1)(0)
D = -4(k^2-1)(0)
D = 0

Similar to Case 1, the discriminant is zero, which means that this equation will always have at least one real root. Therefore, there are no values of k for which this equation has no real roots.

Conclusion:
After analyzing both cases, we can conclude that there are no values of k for which the given equation x^4 (1-2k)x^2 (k^2-1) =0 has no real roots.

For no real root, b²-4ac is less than 0
=> 1+4k²-4k-4k²+4 is less than 0
=> 5-4k is less than 0
=> k>5/4 or k€(5/4, infinity)

For these values of k, x² is not real and so x is also not real.