Nitin Pathak

Given : distance btween two poles = 12 m

Small pole height = 6 m

Big pole height = 11 m

To find : distnace btween two poles

Small to top pole distance can be find by pythgores therom top (hypotenuse)²=(base)²+(height)²

Base =12 m

Height = 11-6= 5 m

(Hypotenuse)²=(12)²+(5)²

Hypotenuse =√(12)²+(5)²m

Hypotenuse=√144+25= √169m

distance between their tops is13 m.

1) 1⁄√2 

Rationalise this number 1 *√2⁄√2 *√2 

= √2/2 

√2 is irrational number and 2 is rational number division of rational number by irrational number is irrational . Hence  1/√2 is an irrational number.

2)7√5   

7 is rational number √5 is an irrational number product of rarional and irrational number is irrational number.

3)6+√5 

6 is a rational number √5 is irrational number sum of rational and irrational number is an irrational number.

To prove : 3+2√5 is rational number.

Ans: 3 is rational number.

2 is rational number.

2√5 is irrational number (product of rational and irrational is an irrational number)

3+2√5 is an irrational number (sum of rational and irrational number is irrational)

Or 

Let P = 3+2√5 is rational number ( we are assuming )

P= 3+2√5

P-3 =2√5

(P-3)÷2=√5

L.H.S. we assume P is a rational number .

P-3 is also rational number substraction of rational number always rational number .

(P-3)÷2 rational number (P-3) divide by rational number 2 hence (P-3)÷2 is rational number .

L.H.S. is rational While R.H.S. √5 is irrational L.H.S ≠R.H.S.

Our assumption is wrong so we can say 3+2√5 is an irrational number.