Problem Solving

M = square root(4) + cube root(4) + fourth root(4) = 2 + cube root(4) + fourth root(4)

Now, for any real number greater than 1, it's n-th root is always greater than 1. Why is so?

Say, b is the n-th root of a.
Therefore, a = b*b* ... *b (n times)

• (1) if b < 1; b*b is always smaller than b. Thus b*b* ... *b (n times) is smaller than b. Hence, a < 1.
  (2) if b > 1; b*b is always greater than b. Thus b*b* ... *b (n times) is greater than b. Hence, a > 1.

These relations are true in the inverse manner also.

Therefore,

• (1) cube root(4) > 1
  (2) fourth root(4) > 1

Thus, M = 2 + cube root(4) + fourth root(4) > 2 + 1 +1 = 4

The correct answer is E.