Math, asked by foryoutobemw, 1 month ago

Prove that V3 + V7
is irrational.​

Answers

Answered by xXMarziyaXx
11

Given:

  • 3 + 5√7

To Prove:

  • 3 + 5√7 is an irrational number.

Proof:

  • Let us assume, to the contrary ,that 3 + 5√7 is rational.

Then, there exists co-primes a and b ( b≠0 ) such that

➽ 3 + 5√7 = a/b

➽ 5√7 = a/b – 3

➽ 5√7 = a – 3b/b

➽ √7 = a – 3b/5b

  • Since, a and b are integers , so (a – 3b/5b) is rational.

  • This √7 is also rational.

  • But this contradicts the fact that √7 is irrational. So, our assumption is wrong.

Hence, (3 + 5√7) is irrational.

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Answered by karanmeena9050
0

Answer:

We have to prove that 3+

7

is irrational.

Let us assume the opposite, that 3+

7

is rational.

Hence 3+

7

can be written in the form

b

a

where a and b are co-prime and b

=0

Hence 3+

7

=

b

a

7

=

b

a

−3

7

=

b

a−3b

where

7

is irrational and

b

a−3b

is rational.

Since,rational

= irrational.

This is a contradiction.

∴ Our assumption is incorrect.

Hence 3+

7

is irrational.

Hence proved.

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