Question

Given that √6 is irrational, and then prove that 5 + 2√6 is also irrational.

Answer 1

SOLUTION

GIVEN

√6 is irrational,

TO PROVE

5 + 2√6 is also irrational.

EVALUATION

If possible let 5 + 2√6 is also rational

$\displaystyle \sf{ \implies \: \frac{1}{5 + 2 \sqrt{6} } \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: \frac{(5 - 2 \sqrt{6} )}{(5 + 2 \sqrt{6} )(5 - 2 \sqrt{6} )} \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: \frac{(5 - 2 \sqrt{6} )}{ {(5)}^{2} - {(2 \sqrt{6} )}^{2} } \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: \frac{(5 - 2 \sqrt{6} )}{25 - 24} \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: (5 - 2 \sqrt{6} ) \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: - ( 5 - 2 \sqrt{6} ) \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: - 5 + 2 \sqrt{6} \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: (5 + 2 \sqrt{6} ) + ( - 5 + 2 \sqrt{6}) \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: 4 \sqrt{6} \: \: is \: \: rational }$

$\displaystyle \sf{ \implies \: \sqrt{6} \: \: is \: \: rational }$

Which is contradiction to be given that √6 is irrational,

Hence 5 + 2√6 is also irrational

Hence proved

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. sum of rational numbers whose absolute value is 7/3

https://brainly.in/question/30899277

2. Write 18.777. . . in p/q form.

https://brainly.in/question/29915810