(i) 2​1​Let us assume 2​1​ is rational.So we can write this number as2​1​=ba​ ---- (1)Here, a and b are two co-prime numbers and b is not equal to zero.Simplify the equation (1) multiply by 2​ both sides, we get1=ba2​​Now, divide by b, we getb=a2​ or ab​=2​Here, a and b are integers so, ab​ is a rational number, so 2​ should be a rational number.But 2​ is a irrational number, so it is contradictory.Therefore, 2​1​ is irrational number.(ii) 75​Let us assume 75​ is rational.So, we can write this number as75​=ba​ ---- (1)Here, a and b are two co-prime numbers and b is not equal to zero.Simplify the equation (1) divide by 7 both sides, we get5​=7ba​Here, a and b are integers, so 7ba​ is a rational number, so 5​ should be a rational number.But 5​ is a irrational number, so it is contradictory.Therefore, 75​ is irrational number.(iii) 6+2​Let us assume 6+2​ is rational.So we can write this number as6+2​=ba​ ---- (1)Here, a and b are two co-prime number and b is not equal to zero.Simplify the equation (1) subtract 6 on both sides, we get2​=ba​−62​=ba−6b​Here, a and b are integers so, ba−6b​ is a rational number, so 2​ should be a rational number.But 2​ is a irrational number, so it is contradictory.Therefore, 6+2​ is irrational number.