Question

Prove that 5 + √3 is Irrational​

Answer 1

Answer:

Step-by-step explanation:

let us assume that 5 + √3 is rational number so we can find two integers a , b. Where a and b are two co - primes number.

= 5 + √3 = a/b

= √3= a/b - 5  

=> a and b are integers so ( a/b -5 ) is rational

But √3 is irrational ( we know that and it is given)

So it arise contradiction due to our wrong assumption that 5 + √3 is rational number.

Hence, 5 +√3 is irrational number.

Hope it will help you ✌️

Answer 2

Answer:

$5+\sqrt{3}\:is\:irrational$

Step-by-step explanation:

$Let\:us\:assume\:that\:5+\sqrt{3}\:is\:rational$

$5+\sqrt{3}=\frac{a}{b}[Where\:a\:and\:b\:are\:co-primes,b\neq 0]$

     $\sqrt{3}=\frac{a}{b}-5$

     $\sqrt{3}=\frac{a-5b}{b}$

$As\:\frac{a-5b}{b}\:is\:rational,\sqrt{3}\:also\:a\:rational\:number.$

$But\:we\:we\:know\:that\:\sqrt{3}\:is\:irrational$.

$This\:contradiction\:has\:arisen\:because\:of\:our\:incorrect\:assumption\:that\:5+\sqrt{3}\\is\:irrational.$

$\therefore{5+\sqrt{3}\:is\:irrational$

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