Less than 4/3

XYYXF · 2008-12-02, 21:50

How to prove without a calculator that

$\sqrt[12]{2}\sqrt[7]{5}$

is less than 4/3?

Uncwilly · 2008-12-02, 23:02

Quote:

Originally Posted by XYYXF

How to prove without a calculator that

Paper and pencil and successive aproximation.

Batalov · 2008-12-03, 02:48

take 84-th power of both sides:
374141099455063118393835625834907722052531250000000 <
374144419156711147060143317175368453031918731001856 :-)

cheesehead · 2008-12-03, 03:31

Quote:

Originally Posted by Batalov

374144419156711147060143317175368453031918731001856 :-)

The 84th power of 4/3 is an even integer?

Oh, wait -- there's a 3^84 factor in the other one, isn't there? Let's see if Factoris (http://wims.unice.fr/wims/wims.cgi) handles that.

Yep: "374141099455063118393835625834907722052531250000000 = 2⁷ × 3⁸⁴ × 5¹²"

davieddy · 2008-12-03, 13:03

I find it difficult to believe a calculator has not been used
in this proof.

2008-12-02, 21:50	#1
XYYXF Jan 2005 Minsk, Belarus 2⁴×5² Posts	Less than 4/3 How to prove without a calculator that $\sqrt[12]{2}\sqrt[7]{5}$ is less than 4/3?

2008-12-03, 02:48	#3
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 9,497 Posts	take 84-th power of both sides: 374141099455063118393835625834907722052531250000000 < 374144419156711147060143317175368453031918731001856 :-) Last fiddled with by Batalov on 2008-12-03 at 02:52

2008-12-03, 13:03	#5
davieddy "Lucan" Dec 2006 England 14512₈ Posts	I find it difficult to believe a calculator has not been used in this proof.