Question #292875

Prove that √(1+√3) is irrational, assuming that √3 is irrational

1
Expert's answer
2022-02-02T10:51:35-0500

Solution:

Given, 3\sqrt3 is irrational.

To prove that 1+3\sqrt{1+\sqrt3} is irrational.

Proof:

Assume 1+3\sqrt{1+\sqrt3} is rational, so that 1+3=ab\sqrt{1+\sqrt3}=\dfrac ab, where a,b are co-primes and b0b\ne0 .

1+3=a2b23=a2b2+1\Rightarrow 1+\sqrt3=\dfrac {a^2}{b^2} \\ \Rightarrow \sqrt3=\dfrac {a^2}{b^2}+1

On left side, 3\sqrt3 is irrational as it is given.

On right side, a2b2+1\dfrac {a^2}{b^2}+1 must be rational because ab\dfrac ab is rational by assumption.

Now, an irrational cannot be equal to rational.

We get a contradiction.

So, our assumption is wrong.

Thus,1+3\sqrt{1+\sqrt3} is irrational.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS