Question

prove that √ 7 is an irrational and hence prove that 3 + 5 √7 is an irrational​

Answer 1

Let assume √7 is rational number

√7=a/b. Where a and b are co prime and their HCF is 1

√7b=a

Square both side

7b²=a²

a² is a factor of 7

a is also a factor of 7 _____________(by theorm)

a=7c__________________________(some integer)

square both side

a²=49c²

7b²=49c²_______________________(a²is 7b²)

b²=7c²

b²is a factor of 7

b is also a factor of 7_______________ (by theorm)

a and b both are factor of 7 and are not co prime number

Which contradict our assumption that√7 is rational number

Therefore√7 is irrational number

Let assume 3+5√7 is a rational number

3+5√7=a/b where a and b are co prime number and their HCF is 1

√7=a/b-3/5

Here rhs is rational number but lhs is also be a rational number but lhs is irrational number

Hence 3+5√7 is irrational number

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