Question 1043011
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Prove that 7+3root2 is not a rational number.
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<pre>
Let assume that {{{7+3sqrt(2)}}} is a rational number.   (1)

Then  {{{7+3sqrt(2)}}} = {{{m/n}}}, where m and n are integer numbers.

Then {{{sqrt(2)}}} = {{{(1/3)*(m/n-7)}}} = {{{(m-7n)/(3n)}}} is the rational number.

But {{sqrt(2)}} is an irrational number  (well known fact.
                                          For the proof see the lesson <A HREF=https://www.algebra.com/algebra/homework/Number-Line/Proving-irrationality-of-some-real-numbers.lesson>Proving irrationality of some real numbers</A> in this site). 

Contradiction.

You got the contradiction.

Hence, the original assumption (1) is wrong.

It implies that  {{{7+3sqrt(2)}}}  is an irrational number.

The proof is completed.
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Solved.