Question

Prove that 2√ 3 + 5 √2 is an irratinal number.​

Answer 1

Step-by-step explanation:

Given:-

2√3 + 5√2

To find:-

Prove that 2√ 3 + 5 √2 is an irratinal number.

Solution:-

Let us assume that

2√3+5√2 is a rational number

It must be in the form of p/q

Where p and q are integers and q≠0

Let 2√3+5√2 = a/b

=> 2√3 = (a/b)-5√2

On squaring both sides then

=> (2√3)^2 = [(a/b)-5√2]^2

=>(2√3)^2 = [(a-5√2b)/b]^2

We know that

(a-b)^2 = a^2-2ab+b^2

=> 12 =[ a^2-2(a)(5√2b)+(5√2b)^2]/b^2

=> 12 b^2 = a^2-10√2 ab+50b^2

=> 10√2 ab = 50b^2-12b^2-a^2

=> 10√2 ab = 38b^2-a^2

=> √2 = (38b^2-a^2)/(10ab)

=>√2 is in the form of p/q

=> √2 is a rational number.

But √2 is not a rational number

It is an irrational number.

This contradicts to our assumption.

2√ 3 + 5 √2 is not a rational number.

2√ 3 + 5 √2 is an irrational number.

Hence, Proved.

Used formula:-

• (a-b)^2 = a^2-2ab+b^2

• The numbers are in the form of p/q where p and q are integers and q≠0 called rational numbers.

• The numbers are not in the form of p/q where p and q are integers and q≠0 called irrational numbers.

Used Method:-

• Indirect method or Method of Contradiction