Question

let a b c d be positive rational number such that a+rootb=c+rootd then either a=c and b=d or b and c are square of rationals. QUICK!!!!​

Answer 1

I think you say to prove it

given

$a + \sqrt{b} = c + \sqrt{d}$

case 1

let a = c

a+√b= c+√d becomes

√b = √d

therefore b=d

case 2.

let a not equal to c

let us take a= c+k,where k is a ration no not equal to zero.

then,

a+√b= c+√d

(c+k)+√b= c+√d

k+√b = √d

squaring on both sides

$k {}^{2} + b + 2k \sqrt{b} = d$

then ,

$2k \sqrt{b} = d - k {}^{2} - b$

note here that RHS is a rational no.

hence √b is a rational no

this is possible only when b us square if a rational no

then ,d is also square of a rational no as k+√b= √d

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Answer 2

hope its help

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