How do you calculate (manually) 2 to the power 3.5?

davee · 11-04-2004, 04:08 AM

Ok - brain getting rusty here.

2^2 = 2 x 2 = 4
2^3 = 2 x 2 x 2 = 8
2^4 = 2 x 2 x 2 x 2 = 16
etc.

How do I calculate 2^3.5? Don't need to know for any reason; it just started bugging me...!

Dave

vharishankar · 11-04-2004, 04:18 AM

We were taught how to do this at school.

Apparently the only way to calculate the above is to use calculus. I forget the exact method, but somebody more in touch with Mathematics might be able to help you.

davee · 11-04-2004, 04:25 AM

I'll maybe do a little research then; it seems like something that should be easy to work out...

Dave

bakery2k · 11-04-2004, 05:22 AM

2^(3.5) = 2^(7*0.5) = (2^7)^(0.5) = 128^(0.5) = sqrt(128) = sqrt(64*2) = sqrt(64)*sqrt(2)

= 8 * sqrt(2)

vharishankar · 11-04-2004, 05:51 AM

Quote:

Originally posted by bakery2k
2^(3.5) = 2^(7*0.5) = (2^7)^(0.5) = 128^(0.5) = sqrt(128) = sqrt(64*2) = sqrt(64)*sqrt(2)

= 8 * sqrt(2)

Perfect mate! You got it. I was just thinking about it and was about to post this simple solution when I saw that somebody had beaten me to it. Thanks...

Mathiasdm · 11-05-2004, 01:25 AM

Or slightly different: 2^(7/2) = 2^(7*1/2) = (2^7)^(1/2) = sqrt (128) = sqrt (64*2) = 8 sqrt 2

Only minor difference ;-)

Poetics · 11-05-2004, 02:00 AM

Blast this finding threads too late to contribute anything extra

Nicely done, gents

kersten78 · 11-05-2004, 02:10 AM

Much easier by addition of exponents. Since (a^x)*(a^y) = a^(x+y),
2^(3.5)=(2^3)*(2^.5), or 8*Sqrt[2]. Same concept as above, just using addition rule instead of multiplication rule. Sorry, I had to post, I'm a math nut.

SciYro · 11-05-2004, 02:24 AM

Quote:

2^(3.5) = 2^(7*0.5) = (2^7)^(0.5) = 128^(0.5) = sqrt(128) = sqrt(64*2) = sqrt(64)*sqrt(2)

= 8 * sqrt(2)

... umm, since i never look calculus I'm a bit behind here .. but what happened to the "0.5" .. suddenly it vanished into 128 without doing a thing... i say the answer seems more like 12 if you count half's

nuka_t · 11-05-2004, 02:37 AM

Quote:

Originally posted by Harishankar
Perfect mate! You got it. I was just thinking about it and was about to post this simple solution when I saw that somebody had beaten me to it. Thanks...

simple???

<---needs to take calculus.

vharishankar · 11-05-2004, 02:56 AM

The above is not the calculus method, everybody. It's simple math

The calculus method is much tougher and is used by students specifically studying calculus. Not used by people who just want the answer.

Explanation

2^{3.5} = 2^{7 * 0.5} = (2^7)^{0.5) = 128^{0.5} (because 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128)

Those who want the answer quickly usually use a calculator, not calculus

Sqrt is another way of saying square root which is same as raised to the power half.

x^{0.5) = sqrt{x}

kersten78 · 11-05-2004, 11:22 AM

What would the caculus method be? It seems like a basic algebra question regarding exponent rules... unless you want to find an approximation. Then maybe you could set up the continued fractions to approximate Sqrt[2], which still seems like an algebra problem, unless you consider some sort of limiting situation...which isn't really going to help since Sqrt[2] is irrational.
Hmm...maybe I missed the calculus of exponents. Who knows...