Question

Show that $5- 2{\sqrt{3}}$ is an irrational number

Answer 1

SOLUTION :  

Let us assume , to the contrary ,that 5 - 2√3 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

5 - 2√3 = a/b

5 - a/b = 2√3

[(5b - a)/b]/2 = √3

(5b - a)/2b  = √3

since, a & b is an integer so, (5b - a)/2b   is a rational number.  

∴ √3 is rational  

But this contradicts the fact that √3 is an irrational number .

Hence, 5 - 2√3 is an irrational .

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Answer 2

hello....

Let 5 - 2√3 is rational. . . ! ! !

therefore;
✔️ 5 - 2√3 = a/b

✔️5 - a/b = 2√3

✔️(5b - a)/b÷/2 = √3

✔️(5b - a)÷2b  = √3


∴ √3 is rational  

But this contradicts that √3 is an irrational number.....

therefore , 5 - 2√3 is an irrational .


thank you.........