Question

Prove that 5 + √8 irritational number

Answer 1

please find the attachment

Answer 2

$Here \: is \: the \: answer \: of \: your \: question$

# Prove = 5 + √8 irrational number

→ Let us assume that 5 + √8 is a rational number.

Now,

5 + √8 = (a ÷ b)

[Here a and b are co-prime numbers]

√8 = (a ÷ b) - 5

√8 = [(a - 5b) ÷ b]

Here, {(a - 5b) ÷ b} is a rational number.

But we know that √8 is a irrational number.

So, {(a - 5b) ÷ b} is also a irrational number.

So, our assumption is wrong.

5 + √8 is a irrational number.

Hence, proved.

__________________________________

How √8 is a irrational number.?

→ √8 = a ÷ b [a and b are co-prime numbers]

b√8 = a

Now, squaring on both side we get,

8b² = a² ........(1)

b² = a² ÷ 8

Here 8 divide a²

and 8 divide a also

Now,

a = 8c [Here c is any integer]

Squaring on both side

a² = 64c²

8b² = 64c² [From (1)]

b² = 8c²

c² = b² ÷ 8

Here 8 divide b²

and 8 divide b also

→ a and b both are co-prime numbers and 8 divide both of them.

So, √8 is a irrational number.

Hence, proved