Prove that `3+2sqrt(5)` is irrational.

Answer

N/a

Explanation

Let us assume that `3+2sqrt(5)` is rational number.
`So," "3+2sqrt(5)=(p)/(q),p,qinZ,q!=0`
(in simplest form i.e., no factor other 1 is common to both p and q)
`rArr" " 2sqrt(5)=(p)/(q)-3rArr" "sqrt(5)=(p-3q)/(2q)" ". . .(1)`
"R.H.S of (1) : Since p and q are integers"
`:." ""p-3qinZand2qinZ`
`So,(p-3q)/(2q)"is rational"`
`rArr"R.H.S. of (1) is rational"` . . .(2)
`"L.H.S. of (1) : Let us assume that"sqrt(5)"is a rational number"` . . .(A)
`"So,"" "sqrt(5)=(a)/(b),a,binZ,b!=`
(in simplest form, i.e., no factor 1 is common to both a and b)
`rArr" "a=sqrt(5b)`
Squaring on both sides, we get
`a^(2)=5b^(2)`
`"But 5 divides"5b^(2)" ". . .(3)`
`rArr" ""5 divides"a^(2)` [from (3)]
`rArr" ""5 divides a"" "((because"5 divides 25"rArr"5 divides 5 also"),("5 divides 100" rArr "5 divides 10 also ect".))`
`"So, let"a=5m,minZ` . . .(4)
`rArr" "a_(2)=25m^(2)" ". . .(4)`
`rArr" "5b^(2)=25m^(2)" "("on squaring")`
`rArr" "b^(2)=5m^(2)" ". . .(5)`
`"But 5 divides"5m^(2)" "`
`rArr" ""5 divides"b^(2)`
`rArr" ""5 divides b"`
`"Let"b=5n,ninZ` . . .(6)
From (4) and (6), we see that 5 divides a nad b both. But we have already assumed that a and b have no common factor other than 1.
So, this contradicts the assumption of (A), that `sqrt(5)` is a rational number.
It means `sqrt(5)` is an irrational number
`rArr` L.H.S. of (1) is irrational. . . .(7)
`:.` From (2) and (7)
`"L.H.S."!="R.H.S."`
Therefore, our supposition that `3+2sqrt(5)` is a rational number, is wrong.
Hence, `3+2sqrt(5)` is an irrational number.