Question

5. Show that 3+5V2 is an irrational number.​

Answer 1

3 + 5√2 is irrational number

__________ [PROVE]

Solution:

• Let us assume that 3 + 5√2 is irrational number.

=> 3 + 5√2 = $\dfrac{a}{b}$

Here, a and b are co-prime numbers.

=> 5√2 = $\dfrac{a}{b}$ - 3

=> 5√2 = $\dfrac{a\:-\:3b}{b}$

=> √2 = $\dfrac{a\:-\:3b}{5b}$

Here;

$\dfrac{a\:-\:3b}{5b}$ is rational number.

So, √2 is also a rational number.

But we know that √2 is irrational number.

So, our assumption is wrong.

3 + 5√2 is irrational number.

Hence, proved.

______________________________

Answer 2

Let 3 + 5√2 is irrational number.

3 + 5√2 = a/b

5√2 = a/b - 3

√2 = a - 3b/b

a-3b/b = Rational number

Then √2 is also a rational number.

Bu √2 is irrational number. So 3 + 5√2 is irrational number.