If M = √4 + ∛4 + ∜4, then the value of M is:

A) less than 3
B) equal to 3
C) between 3 and 4
D) equal to 4
E) greater than 4

We are given that M = √4 + ^3√4 + ^4√4. We need to determine the approximate value of M.

Since √4 = 2, we need to determine the value of 2 + ^3√4 + ^4√4

Let's determine the approximate value of ^3√4. To find this value, we need to find the perfect cube roots just below and just above the cube root of 4.

^3√1 < ^3√4 < ^3√8

1 < ^3√4 < 2

Let's next determine the approximate value of ^4√4. To determine this value, we need to find the perfect fourth roots just below and just above the fourth root of 4.

^4√1 < ^4√4 < ^4√16

1 < ^4√4 < 2

Since both ^3√4 and ^4√4 are greater than 1, √4 + ^3√4 + ^4√4 > 2 + 1 + 1 = 4.

Answer: E

√4 = 2

fourth root of 4 = 4^(1/4) = (2^2)^(1/4) = √2

So we've got 2 + √2 + (some number between 2 and √2), which means our sum is

2 + √2 + √2 < our sum < 2 + √2 + 2

The lesser of these (2 + √2 + √2) is ≈ 2 + 1.4 + 1.4, so our sum is greater than 4.