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prob_64.note.1
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sqrt(2)
= 1 + sqrt(2)-1 ==> 1, 1/( sqrt(2)-1 )
1/( sqrt(2)-1 )
= ( sqrt(2)+1 )/1
= 2 + sqrt(2)-1 ==> 2, 1/( sqrt(2)-1 )
sqrt(3)
= 1 + sqrt(3)-1 ==> 1, 1/( sqrt(3)-1 )
1/( sqrt(3)-1 )
= ( sqrt(3)+1 )/2
= 1 + ( sqrt(3)-1 )/2 ==> 1, 2/( sqrt(3)-1 )
2/( sqrt(3)-1 )
= 2*( sqrt(3)+1 )/2
= ( sqrt(3)+1 )/1
= 2 + sqrt(3)-1 ==> 2, 1/( sqrt(3)-1 )
sqrt(5)
= 2 + sqrt(5)-2 ==> 2, 1/( sqrt(5)-2 )
1/( sqrt(5)-2 )
= ( sqrt(5)+2 )/1
= 4 + sqrt(5)-2 ==> 4, 1/( sqrt(5)-2 )
sqrt(6)
= 2 + sqrt(6)-2 ==> 2, 1/( sqrt(6)-2 )
1/( sqrt(6)-2 )
= ( sqrt(6)+2 )/2
= 2 + ( sqrt(6)-2 )/2 ==> 2, 2/( sqrt(6)-2 )
2/( sqrt(6)-2 )
= 2*( sqrt(6)+2 )/2
= ( sqrt(6)+2 )/1
= 4 + sqrt(6)-2 ==> 4, 1/( sqrt(6)-2 )
sqrt(7)
= 2 + sqrt(7)-2 ==> 2, 1/( sqrt(7)-2 )
1/( sqrt(7)-2 )
= ( sqrt(7)+2 )/3
= 1 + ( sqrt(7)-1 )/3 ==> 1, 3/( sqrt(7)-1 )
3/( sqrt(7)-1 )
= 3*( sqrt(7)+1 )/6
= ( sqrt(7)+1 )/2
= 1 + ( sqrt(7)-1 )/2 ==> 1, 2/( sqrt(7)-1 )
2/( sqrt(7)-1 )
= 2*( sqrt(7)+1 )/6
= ( sqrt(7)+1 )/3
= 1 + ( sqrt(7)-2 )/3 ==> 1, 3/( sqrt(7)-2 )
3/( sqrt(7)-2 )
= 3*( sqrt(7)+2 )/3
= ( sqrt(7)+2 )/1
= 4 + ( sqrt(7)-2 )/1 ==> 4, 1/( sqrt(7)-2 )
sqrt(8)
= 2 + sqrt(8)-2 ==> 2, 1/( sqrt(8)-2 )
1/( sqrt(8)-2 )
= ( sqrt(8)+2 )/4
= 1 + ( sqrt(8)-2 )/4 ==> 1, 4/( sqrt(8)-2 )
4/( sqrt(8)-2 )
= 4*( sqrt(8)+2 )4
= ( sqrt(8)+2 )/1
= 4 + sqrt(8)-2 ==> 4, 1/( sqrt(8)-2 )
sqrt(10)
= 3 + sqrt(10)-3 ==> 3, 1/( sqrt(10)-3 )
1/( sqrt(10)-3 )
= ( sqrt(10)+3 )/1
= 6 + sqrt(10)-3 ==> 6, 1/( sqrt(10)-3 )
sqrt(11)
= 3 + sqrt(11)-3 ==> 3, 1/( sqrt(11)-3 )
1/( sqrt(11)-3 )
= ( sqrt(11)+3 )/2
= 3 + ( sqrt(11)-3 )/2 ==> 3, 2/( sqrt(11)-3 )
2/( sqrt(11)-3 )
= 2*( sqrt(11)+3 )/2
= ( sqrt(11)+3 )/1
= 6 + sqrt(11)-3 ==> 6, 1/( sqrt(11)-3 )
sqrt(12)
= 3 + sqrt(12)-3 ==> 3, 1/( sqrt(12)-3 )
1/( sqrt(12)-3 )
= ( sqrt(12)+3 )/3
= 2 + ( sqrt(12)-3 )/3 ==> 2, 3/( sqrt(12)-3 )
3/( sqrt(12)-3 )
= 3*( sqrt(12)+3 )/3
= ( sqrt(12)+3 )/1
= 6 + sqrt(12)-3 ==> 6, 1/( sqrt(12)-3 )
sqrt(13)
= 3 + sqrt(13)-3 ==> 3, 1/( sqrt(13)-3 )
1/( sqrt(13)-3 )
= ( sqrt(13)+3 )/4
= 1 + ( sqrt(13)-1 )/4 ==> 1, 4/( sqrt(13)-1 )
4/( sqrt(13)-1 )
= 4*( sqrt(13)+1 )/12
= ( sqrt(13)+1 )/3
= 1 + ( sqrt(13)-2 )/3 ==> 1, 3/( sqrt(13)-2 )
3/( sqrt(13)-2 )
= 3*( sqrt(13)+2 )/9
= ( sqrt(13)+2 )/3
= 1 + ( sqrt(13)-1 )/3 ==> 1, 3/( sqrt(13)-1 )
3/( sqrt(13)-1 )
= 3*( sqrt(13)+1 )/12
= ( sqrt(13)+1 )/4
= 1 + ( sqrt(13)-3 )/4 ==> 1, 4/( sqrt(13)-3 )
4/( sqrt(13)-3 )
= 4*( sqrt(13)+3 )/4
= ( sqrt(13)+3 )/1
= 6 + sqrt(13)-3 ==> 6, 1/( sqrt(13)-3 )
sqrt(14)
= 3 + sqrt(14)-3 ==> 3, 1/( sqrt(14)-3 )
1/( sqrt(14)-3 )
= ( sqrt(14)+3 )/5
= 1 + ( sqrt(14)-2 )/5 ==> 1, 5/( sqrt(14)-2 )
5/( sqrt(14)-2 )
= 5*( sqrt(14)+2 )/10
= ( sqrt(14)+2 )/2
= 2 + ( sqrt(14)-2 )/2 ==> 2, 2/( sqrt(14)-2 )
2/( sqrt(14)-2 )
= 2*( sqrt(14)+2 )/10
= ( sqrt(14)+2 )/5
= 1 + ( sqrt(14)-3 )/5 ==> 1, 5/( sqrt(14)-3 )
5/( sqrt(14)-3 )
= 5*( sqrt(14)+3 )/5
= ( sqrt(14)+3 )/1
= 6 + sqrt(14)-3 ==> 6, 1/( sqrt(14)-3 )
sqrt(15)
= 3 + sqrt(15)-3 ==> 3, 1/( sqrt(15)-3 )
1/( sqrt(15)-3 )
= ( sqrt(15)+3 )/6
= 1 + ( sqrt(15)-3 )/6 ==> 1, 6/( sqrt(15)-3 )
6/( sqrt(15)-3 )
= 6*( sqrt(15)+3 )/6
= ( sqrt(15)+3 )/1
= 6 + sqrt(15)-3 ==> 6, 1/( sqrt(15)-3 )
sqrt(17)
= 4 + sqrt(17)-4 ==> 4, 1/( sqrt(17)-4 )
1/( sqrt(17)-4 )
= ( sqrt(17)+4 )/1
= 8 + sqrt(17)-4 ==> 8, 1/( sqrt(17)-4 )
sqrt(18)
= 4 + sqrt(18)-4 ==> 4, 1/( sqrt(18)-4 )
1/( sqrt(18)-4 )
= ( sqrt(18)+4 )/2
= 4 + ( sqrt(18)-4 )/2 ==> 4, 2/( sqrt(18)-4 )
2/( sqrt(18)-4 )
= 2*( sqrt(18)+4 )/2
= ( sqrt(18)+4 )/1
= 8 + sqrt(18)-4 ==> 8, 1/( sqrt(18)-4 )
sqrt(19)
= 4 + sqrt(19)-4 ==> 4, 1/( sqrt(19)-4 )
1/( sqrt(19)-4 )
= ( sqrt(19)+4 )/3
= 2 + ( sqrt(19)-2 )/3 ==> 2, 3/( sqrt(19)-2 )
3/( sqrt(19)-2 )
= 3*( sqrt(19)+2 )/15
= ( sqrt(19)+2 )/5
= 1 + ( sqrt(19)-3 )/5 ==> 1, 5/( sqrt(19)-3 )
5/( sqrt(19)-3 )
= 5*( sqrt(19)+3 )/10
= ( sqrt(19)+3 )/2
= 3 + ( sqrt(19)-3 )/2 ==> 3, 2/( sqrt(19)-3 )
2/( sqrt(19)-3 )
= 2*( sqrt(19)+3 )/10
= ( sqrt(19)+3 )/5
= 1 + ( sqrt(19)-2 )/5 ==> 1, 5/( sqrt(19)-2 )
5/( sqrt(19)-2 )
= 5*( sqrt(19)+2 )/15
= ( sqrt(19)+2 )/3
= 2 + ( sqrt(19)-4 )/3 ==> 2, 3/( sqrt(19)-4 )
3/( sqrt(19)-4 )
= 3*( sqrt(19)+4 )/3
= ( sqrt(19)+4 )/1
= 8 + sqrt(19)-4 ==> 8, 1/( sqrt(19)-4 )
sqrt(20)
= 4 + sqrt(20)-4 ==> 4, 1/( sqrt(20)-4 )
1/( sqrt(20)-4 )
= ( sqrt(20)+4 )/4
= 2 + ( sqrt(20)-4 )/4 ==> 2, 4/( sqrt(20)-4 )
4/( sqrt(20)-4 )
= 4*( sqrt(20)+4 )/4
= ( sqrt(20)+4 )/1
= 8 + sqrt(20)-4 ==> 8, 1/( sqrt(20)-4 )
sqrt(21)
= 4 + sqrt(21)-4 ==> 4, 1/( sqrt(21)-4 )
1/( sqrt(21)-4 )
= ( sqrt(21)+4 )/5
= 1 + ( sqrt(21)-1 )/5 ==> 1, 5/( sqrt(21)-1 )
5/( sqrt(21)-1 )
= 5*( sqrt(21)+1 )/20
= ( sqrt(21)+1 )/4
= 1 + ( sqrt(21)-3 )/4 ==> 1, 4/( sqrt(21)-3 )
4/( sqrt(21)-3 )
= 4*( sqrt(21)+3 )/12
= ( sqrt(21)+3 )/3
= 2 + ( sqrt(21)-3 )/3 ==> 2, 3/( sqrt(21)-3 )
3/( sqrt(21)-3 )
= 3*( sqrt(21)+3 )/12
= ( sqrt(21)+3 )/4
= 1 + ( sqrt(21)-1 )/4 ==> 1, 4/( sqrt(21)-1 )
4/( sqrt(21)-1 )
= 4*( sqrt(21)+1 )/20
= ( sqrt(21)+1 )/5
= 1 + ( sqrt(21)-4 )/5 ==> 1, 5/( sqrt(21)-4 )
5/( sqrt(21)-4 )
= 5*( sqrt(21)+4 )/5
= ( sqrt(21)+4 )/1
= 8 + sqrt(21)-4 ==> 8, 1/( sqrt(21)-4 )
sqrt(22)
= 4 + sqrt(22)-4 ==> 4, 1/( sqrt(22)-4 )
1/( sqrt(22)-4 )
= ( sqrt(22)+4 )/6
= 1 + ( sqrt(22)-2 )/6 ==> 1, 6/( sqrt(22)-2 )
6/( sqrt(22)-2 )
= 6*( sqrt(22)+2 )/18
= ( sqrt(22)+2 )/3
= 2 + ( sqrt(22)-4 )/3 ==> 2, 3/( sqrt(22)-4 )
3/( sqrt(22)-4 )
= 3*( sqrt(22)+4 )/6
= ( sqrt(22)+4 )/2
= 4 + ( sqrt(22)-4 )/2 ==> 4, 2/( sqrt(22)-4 )
2/( sqrt(22)-4 )
= 2*( sqrt(22)+4 )/6
= ( sqrt(22)+4 )/3
= 2 + ( sqrt(22)-2 )/3 ==> 2, 3/( sqrt(22)-2 )
3/( sqrt(22)-2 )
= 3*( sqrt(22)+2 )/18
= ( sqrt(22)+2 )/6
= 1 + ( sqrt(22)-4 )/6 ==> 1, 6/( sqrt(22)-4 )
6/( sqrt(22)-4 )
= 6*( sqrt(22)+4 )/6
= ( sqrt(22)+4 )/1
= 8 + sqrt(22)-4 ==> 8, 1/( sqrt(22)-4 )
sqrt(23)
= 4 + sqrt(23)-4 ==> 4, 1/( sqrt(23)-4 )
1/( sqrt(23)-4 )
= ( sqrt(23)+4 )/7
= 1 + ( sqrt(23)-3 )/7 ==> 1, 7/( sqrt(23)-3 )
7/( sqrt(23)-3 )
= 7*( sqrt(23)+3 )/14
= ( sqrt(23)+3 )/2
= 3 + ( sqrt(23)-3 )/2 ==> 3, 2/( sqrt(23)-3 )
2/( sqrt(23)-3 )
= 2*( sqrt(23)+3 )/14
= ( sqrt(23)+3 )/7
= 1 + ( sqrt(23)-4 )/7 ==> 1, 7/( sqrt(23)-4 )
7/( sqrt(23)-4 )
= 7*( sqrt(23)+4 )/7
= ( sqrt(23)+4 )/1
= 8 + sqrt(23)-4 ==> 8, 1/( sqrt(23)-4 )
sqrt(24)
= 4 + sqrt(24)-4 ==> 4, 1/( sqrt(24)-4 )
1/( sqrt(24)-4 )
= ( sqrt(24)+4 )/8
= 1 + ( sqrt(24)-4 )/8 ==> 1, 8/( sqrt(24)-4 )
8/( sqrt(24)-4 )
= 8*( sqrt(24)+4 )/8
= ( sqrt(24)+4 )/1
= 8 + sqrt(24)-4 ==> 8, 1/( sqrt(24)-4 )