(Q3).Prove that (2 √3+ √5) is an irrational number. Also check whether (2 √3 + √5)(2√3 - √5) is rational or irrational.

Rational Number is defined as the number which can be written in a ratio of two integers.

An irrational number is a number which cannot be expressed in a ratio of two integers.

Let us assume that 2√3 + √5 is rational number.

Let P= 2√3 + √5 is rational

on squaring both sides we get

P2 = (2√3 + √5)2 = (2√3)2 + (√5)2+2 × 2√3 × √5

P2 = 12 + 5 + 4√15

P2 = 17 + 4√15

P2 - 17
4
 = √15..........(1)
Since P is rational no. therefore
P2 - 17
4
 is also rational.

But √15 is irrational in equation(1)

P2 - 17
4
 = √15

Rational ≠ irrational

P = (2 - √3 + √5)(2 √3 - √5)

P = (2√3)2 - (√5)2

P = 12 -

P =

Hence P is Rational as
p
q
 =
7
1
 & both p & q are numbers.