Question

Prove that 2+3√3 is irrational number if it is given that √3 is irrational number​

Answer 1

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Answer 2

$\bf\huge\red{Answer:-}$

Given:

3+2√3

To Find :

3+2√3 is an irrational number

Solution :

First, shall we prove that √3 is an irrational number.

So,

Let us assume that√3 is an rational number, a and b are co prime numbers and b ≠ 0.

√3 = a/b

b√5 = a

Square on both sides:

(b√3)² = a²

3b²= a² -------(★)

b²=a²/3

•°• 3 is an prime number

If 3 divides a² then 3 divides a also

3 is a factor of a ------(1)

now,

assume that a = 3c in (★)

3b²=(3c)²

3b²=9c²

c²=b²/3

b² is divisible by 3

•°• 3 is a factor of b -------(2)

From (1) and (2)

a and b are not co prime number because HCF is 1.

So, a and b is not a rational number, our assumption is wrong.

So, √3 is irrational number.

now