Rational or irrational number?

[MFACCT-001]

10 raised to (3/7) power by calculator yields a non-repeating, 15-digit (calculator limit) result: is this a rational or irrational number?

 

[tkhunny]

15 digits is a very small sample. How do you know it is non-repeating past that?

Here's some more: 2.6826957952797257476988026806276

Still doesn't answer the question. Can you express 10^(3/7) as a ratio of two integers?

[math]10^{3/7} = \dfrac{a}{b}\implies 10 = \left(\dfrac{a}{b}\right)^{2}\cdot \left(\dfrac{a}{b}\right)^{1/3}[/math]. Does this get us anywhere?

 

[MFACCT-001]

I was hoping that someone could verify that 10 raised to the (3/7) power is (or is not) a rational number. Your investigation (nice work) extended the calculated result to (31) decimals, showing a non-ending condition or repeating pattern. Perhaps there's a proof or other method to verify the computed result is rational (or irrational).

 

[pka]

I was hoping that someone could verify that 10 raised to the (3/7) power is (or is not) a rational number. Your investigation (nice work) extended the calculated result to (31) decimals, showing a non-ending condition or repeating pattern. Perhaps there's a proof or other method to verify the computed result is rational (or irrational).

SEE HERE

 

[Dr.Peterson]

I was hoping that someone could verify that 10 raised to the (3/7) power is (or is not) a rational number. Your investigation (nice work) extended the calculated result to (31) decimals, showing a non-ending condition or repeating pattern. Perhaps there's a proof or other method to verify the computed result is rational (or irrational).

I believe tkhunny was suggesting the beginning of a way to prove that it is or is not irrational, similar to the proof that the square root of 2 is irrational. At the same time, that suggests that it is at least very likely that this, too, is irrational. To make that definite, you can either do the proof, or look it up (like pka), or look for a general theorem that applies.

The 31 decimal places accomplishes nothing at all, actually! It takes all the decimal places (that is, the exact value) in order to say whether a number is irrational.

 

[pka]

Never having had a course in number theory I have no idea how to prove the number is irrational.
I like analysis proofs like: If \(n\in\mathbb{Z}^+\) that is not a square then \(\sqrt n\) is not rational.

 

[Steven G]

10 raised to (3/7) power by calculator yields a non-repeating, 15-digit (calculator limit) result: is this a rational or irrational number?

10^(3/7) = (10^3)^(1/7) = 1000^(1/7) Is that rational? What have you tried? Where did you get stuck? We are a math help forum where we prefer to help students solve their own problems vs just giving out the answer (which someone did do anyways).

Come back, discuss the problem with us by telling us some ideas you have and we will help you prove that the number is irrational.