Question

prove thae following as irrational number √3+√7​

Answer 1

Step-by-step explanation:

√3+√7 Let a rational no.

equal to p/q where q =0

√3+√7=p/q

√3=p/q-√7

squaring both sides

p^2/q^2+7-2p/q√7=3

p^2/q^2+4=2p/q√7

p/q+4=2√7

p/q+2=√7

p/q +2 is a rational number

√7 is a irrational number

a rational number and a irrational number can never be equal so our supposition is wrong

Hence proved that √3+√7 is a irrational number

Answer 2

Let us assume the opposite, that 3+√7 is rational.

Hence, 3+√7 can be written in the form a/b

where a and b are co-prime and b is not equal to 0.

Hence, 3+√7=a/b

= √7=a/b-3

=√7= a-3b/b

where √7 is irrational and a-3b/b is rational.

Since, rational is not equal to irrational.

This is a contradiction.

Our assumption is incorrect.

Hence, 3+√7 is irrational.

Hence proved.