Question

Solve (2+√3) is irrational

Answer 1

Answer:

yes, 2+ $\sqrt{3}$is a irrational number.

Step-by-step explanation:

let's first prove that  $\sqrt{3}$ is a irrational number so that if  $\sqrt{3}$ is a irrational number then 2+ $\sqrt{3}$ is also irrational number.

To prove  $\sqrt{3}$ is a irrational number

Let  $\sqrt{3}$ be rational number.

Q=$\frac{a}{b}$. where a and b are integer numbers and b$\neq 0$.

so  $\sqrt{3}$=$\frac{a}{b}$

(squaring on both sides)

$(\sqrt{3}) ^{2} =(\frac{a}{b})^{2}$

3=$\frac{a^{2} }{b^{2} }$

$a^{2} =3b^{2}$ ...........(i) equation

now taking a=3c

(squaring on both sides)

$(a)^{2} =(3c)^{2}$

$a^{2} =9 c^{2}$

(now substituting the value of $a^{2}$ from (i) equation)

3$b^{2}$=9$c^{2}$ ........(ii) equation

since rational numbers are co-prime number a and b will not have any common factor

as 3 and 9 are common factors from (ii) equation, $\sqrt{3}$ is not a rational number . it is irrational number.

Therefore, as $\sqrt{3}$ is an irrational number even 2+$\sqrt{3}$ is also an irrational number.

hope so it helped you.

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