Đáp án:
`a)` `29 + 12 sqrt(5) = (sqrt(20)+sqrt9)^2;`
`b)` `49 + 8 sqrt3 = (sqrt(48)+1)^2`
`c)` `14 - 6 sqrt3 = (sqrt(7 + sqrt(22))-sqrt(7 - sqrt(22)))^2;`
`d)` `23 - 8 sqrt7=(sqrt(16)+sqrt(7))^2`
`e)` `15 - 4 sqrt11 = (sqrt(11) + 2)^2;`
`g)` `28 - 10sqrt5 = (sqrt(14+sqrt(71))+sqrt(14-sqrt(71)))^2`
`h)` `17 - 12 sqrt2 = (3+sqrt8)^2;i)` `43 - 30 sqrt2 =(5+sqrt(18))^2;`
`k)` `51 - 10 sqrt2 =(sqrt(50)+1)^2; `
`l)` `59 - 12 sqrt5 = (sqrt((59+sqrt(2761))/2)+sqrt((59-sqrt(2761))/2))^2`
Giải thích các bước giải:
`a)`` 29 + 12 sqrt(5) = 20 + 9 + 2 . 6 . sqrt5 = 20 + 2. sqrt(36.5) + 9`
`= sqrt(20)^2 + 2 sqrt(180) + sqrt(9)^2 = sqrt(20)^2 + 2.sqrt(20.9) + sqrt(9)^2`
`= (sqrt(20)+sqrt9)^2`
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`b)`` 49 + 8 sqrt3 = 48 + 1 + 2.4sqrt3 = 48 + 2.sqrt(16.3) + 1`
`= sqrt(48)^2 + 2.sqrt(48).1 + 1^2 = (sqrt(48)+1)^2`
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`c)`` 14 - 6 sqrt3 = 7 + 7 - 2.3.sqrt3 = 7 + 7 + sqrt(22) - sqrt(22) - 2.sqrt(3^(2).3)`
`= (7 + sqrt(22)) - 2.sqrt(27) + (7 - sqrt(22))`
`= sqrt(7 + sqrt(22))^2 - 2.sqrt((7 + sqrt(22))(7 - sqrt(22))) + sqrt(7 - sqrt(22))^2`
`= (sqrt(7 + sqrt(22))-sqrt(7 - sqrt(22)))^2`
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`d)` `23 - 8 sqrt7= 16 + 7 - 2.4.sqrt7 = 16 - 2.sqrt(4^(2).7) + 7`
`= sqrt(16)^2 - 2sqrt(112) + sqrt(7)^2 = sqrt(16)^2 - 2sqrt(16.7) + sqrt(7)^2`
`= (sqrt(16)+sqrt(7))^2`
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`e)` `15 - 4 sqrt11 = 11 + 4 + 2.2sqrt(11) = 11 + 2sqrt(2^(2).11) + 4`
`= sqrt(11)^2 + 2sqrt(44) + sqrt(4)^2 = sqrt(11)^2 + 2sqrt(11.4) + sqrt(4)^2`
`= (sqrt(11) + sqrt4)^2 = (sqrt(11) + 2)^2`
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`g)` `28 - 10sqrt5 = 14 + 14 - 2.5.sqrt5 = 14 + 14 + sqrt(71) - sqrt(71) - 2.sqrt(5^(2).5)`
`= (14+sqrt(71)) - 2.sqrt(125) + (14-sqrt(71))`
`= sqrt(14+sqrt(71))^2 - 2.sqrt((14+sqrt(71))(14-sqrt(71))) + sqrt(14-sqrt(71))^2`
`= (sqrt(14+sqrt(71))+sqrt(14-sqrt(71)))^2`
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`h)` `17 - 12 sqrt2 =9 + 8 - 3.2.2.sqrt2 = 9 - 3.2.sqrt(2^(2).2) + 8`
`= 3^2 - 2.3.sqrt8 + sqrt(8)^2 = (3+sqrt8)^2`
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`i)` `43 - 30 sqrt2 = 25+ 18 - 5.2.3.sqrt2= 25 - 2.5.sqrt(3^(2).2) + 18`
`= 5^2 + 2.5.sqrt(18) + sqrt(18)^2 = (5+sqrt(18))^2`
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`k)` `51 - 10 sqrt2 = 50 + 1 - 2.5.sqrt2 = 50 - 2.sqrt(5^(2).2) + 1`
`= sqrt(50)^2 - 2sqrt(50.1) + 1^2 = (sqrt(50)+1)^2`
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`l)` `59 - 12 sqrt5 = (108)/2 - 2.6.sqrt5 = (59)/2 + (59)/2 - 2.sqrt(6^(2).5) `
`= (59)/2 - 2.sqrt(180) + (59)/2`
`= (59)/2 - 2.sqrt(180) + (59)/2 + (sqrt(2761))/2 - (sqrt(2761))/2`
`= ((59)/2+(sqrt(2761))/2)- 2.sqrt(180) +((59)/2-(sqrt(2761))/2)`
`= (59+sqrt(2761))/2- 2.sqrt(180) + (59-sqrt(2761))/2`
`= sqrt((59+sqrt(2761))/2)^2- 2.sqrt((59+sqrt(2761))/2.(59-sqrt(2761))/2) + sqrt((59-sqrt(2761))/2)^2`
`= (sqrt((59+sqrt(2761))/2)+sqrt((59-sqrt(2761))/2))^2`
