If M = 4^(1/2) + 4^(1/3) + 4^(1/4) , the the value of M is

[gphil]

I am not sure how to approach the problem. Thanks for help!!!


If M = 4^(1/2) + 4^(1/3) + 4^(1/4) , the 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 - correct answer

[Amit]

If M = 4^(1/2) + 4^(1/3) + 4^(1/4) , the the value of M is:

Let's start with what we know:
4^(1/2) = 2
4^(1/4) = root 2 = 1.4 ish.

We also know that 4^(1/3) lies somewhere between 2 and 1.4. Let's

4^(1/2) + 4^(1/4) = 2 + 1.4 = 3.4

Now let's add the MIN value of 4^(1/3)
3.4 + 1.4 = 4.8

Since the sum is at MIN 4.8, it has to be greater than 4.

[gphil]

How did you find value of 4^(1/3)?

[Guest]

The tricky part of the question is that you can't find 4^1/3 on the spot (you could but it would take too much time).

So you start with just M = 4^(1/2) + 4^(1/4) for now and we'll add the rest later.

M = 4^(1/2) + 4^(1/4) + (4^1/3)
= 2 + 1.4ish + (4^1/3)
= 3.4 ish + (4^1/3) <<already we know the answer is not A or B

Since we can't calculate (4^1/3) precisely, we can estimate from what we know.
4^(1/2) = 2
4^(1/4) = 1.4
so we know (4^1/3) lies somewhere BETWEEN 2 and 1.4. (If this is unclear try 4^1/2 with calculator, then 4^1/3, 4^1/4 and you'll see them decreasing)

So since we know (4^1/3) is at LEAST 1.4, if we add that to our current total of
M = 3.4 ish + (4^1/3)
= 3.4 ish + 1.4 ish (at LEAST)
M > 3.4 + 1.4
M > 4.8

So we need not calculate 4^(1/3) since we know E must be the answer!

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

[gphil]

Thanks a lot!!!

[StaceyKoprince]

Excellent - glad to see some of our posters are taking over the instructor's jobs and giving great answers! :)

[BRYNAT]

I may be wrong, but isnt is easier just factoring out 4^1/2?

4^1/2 (1 + 4^1/2 + 4^1/3)

so

2(1+2+ 4^1/3)= more than 4

P.S.: I wanna thank manhattan for having this nice forum open to everyone.

[rfernandez]

I may be wrong, but isnt is easier just factoring out 4^1/2?

4^1/2 (1 + 4^1/2 + 4^1/3)

so

2(1+2+ 4^1/3)= more than 4

P.S.: I wanna thank manhattan for having this nice forum open to everyone.

Actually, your expression 4^1/2 (1 + 4^1/2 + 4^1/3) yields 4^(1/2) + 4 + 4^(5/6) ==> not the same expression we're given at the start.

If you want to go about it by factoring out, the common factor would be 4^(1/4).

You'd get 4^(1/4) [4^(1/4) + 4^(1/12) + 1]. Check to see that this is correct by distributing back through. For this problem, I wouldn't say this method is easier because you have to deal with 4^(1/12).

Factoring out common factors can be tricky when you deal with rational exponents. The technique is no different than when you deal with integer exponents, however. Just pull out the smallest power, as in:

x^3 + x^4 + x^6 = x^3(1 + x + x^3)

Rey