Hướng dẫn trả lời:
a) `7sqrt{a} - 5sqrt{16a^3} + 2sqrt{25ab^2} - 7sqrt{9a}`
`= 7sqrt{a} - 5sqrt{(4a)^2cdota} + 2sqrt{(5b)^2a} - 7sqrt{3^2a}`
`= 7sqrt{a} - 5cdot4|a|sqrt{a} + 2cdot5|b|sqrt{a} - 7cdot3sqrt{a}`
`= 7sqrt{a} - 20|a|sqrt{a} + 10|b|sqrt{a} - 21sqrt{a}`
`= - 20|a|sqrt{a} + 10|b|sqrt{a} + (7sqrt{a} - 21sqrt{a})`
`= - 20|a|sqrt{a} + 10|b|sqrt{a} - 14sqrt{a}`
b) `2sqrt{a} + 5sqrt{a/9} - asqrt{16/a} + sqrt{a^3}`
`= 2sqrt{a} + 5sqrt{a/3^2} - sqrt{a^2cdot4^2/a} + sqrt{a^2cdota}` (Vì `a > 0`)
`= 2sqrt{a} + 5/3sqrt{a} - 4sqrt{a} + asqrt{a}` (Vì `a > 0`)
`= (2sqrt{a} + 5/3sqrt{a} - 4sqrt{a}) + asqrt{a}`
`= {-sqrt{a}}/3 + asqrt{a}`
`= {-sqrt{a}}/3 + {3asqrt{a}}/{3}`
`= {-sqrt{a} + 3asqrt{a}}/{3}`
`= {-sqrt{a}cdot(1 - 3a)}/{3}`
c) `2sqrt{18} + 3sqrt[8} - 8sqrt{32} - sqrt{50}`
`= 2sqrt{9cdot2} + 3sqrt[4cdot2} - 8sqrt{16cdot2} - sqrt{25cdot2}`
`= 2sqrt{3^2cdot2} + 3sqrt[2^2cdot2} - 8sqrt{4^2cdot2} - sqrt{5^2cdot2}`
`= 2cdot3sqrt{2} + 3cdot2sqrt[2} - 8cdot4sqrt{2} - 5sqrt{2}`
`= 6sqrt{2} + 6sqrt[2} - 32sqrt{2} - 5sqrt{2}`
`= - 25sqrt{2}`
d) `(2sqrt{6} - 4sqrt{3} - 5sqrt{2} - 1/4sqrt{48})cdot3sqrt{6}`
`= (2sqrt{6} - 4sqrt{3} - 5sqrt{2} - 1/4sqrt{4^2cdot3})cdot3sqrt{6}`
`= (2sqrt{6} - 4sqrt{3} - 5sqrt{2} - 1/4cdot4sqrt{3})cdot3sqrt{6}`
`= (2sqrt{6} - 4sqrt{3} - 5sqrt{2} - sqrt{3})cdot3sqrt{6}`
`= (2sqrt{6} - 5sqrt{3} - 5sqrt{2})cdot3sqrt{6}`
`= 2sqrt{6}cdot3sqrt{6} - 5sqrt{3}cdot3sqrt{6} - 5sqrt{2}cdot3sqrt{6}`
`= 6cdot(sqrt{6})^2 - 15sqrt{18} - 15sqrt{12}`
`= 6cdot6 - 15sqrt{3^2cdot2} - 15sqrt{2^2cdot3}`
`= 36 - 15cdot3sqrt{2} - 15cdot2sqrt{3}`
`= 36 - 45sqrt{2} - 30sqrt{3}`
i) `({9 - 2sqrt{14}}/{sqrt{7} - sqrt{2}})^2 + ({9 + 2sqrt{14}}/{sqrt{7} + sqrt{2}})^2`
`= ({7 - 2sqrt{14} + 2}/{sqrt{7} - sqrt{2}})^2 + ({7 + 2sqrt{14} + 2}/{sqrt{7} + sqrt{2}})^2`
`= ({(sqrt{7})^2 - 2cdot sqrt{7}cdot sqrt{2} + (sqrt{2})^2}/{sqrt{7} - sqrt{2}})^2 + ({(sqrt{7})^2 + 2cdot sqrt{7}cdot sqrt{2} + (sqrt{2})^2}/{sqrt{7} + sqrt{2}})^2`
`= ({(sqrt{7} - sqrt{2})^2}/{sqrt{7} - sqrt{2}})^2 + ({(sqrt{7} + sqrt{2})^2}/{sqrt{7} + sqrt{2}})^2`
`= (sqrt{7} - sqrt{2})^2 + (sqrt{7} + sqrt{2})^2`
`= [(sqrt{7})^2 - 2cdot sqrt{7}cdot sqrt{2} + (sqrt{2})^2] + [(sqrt{7})^2 + 2cdot sqrt{7}cdot sqrt{2} + (sqrt{2})^2]`
`= (7 - 2sqrt{14} + 2) + (7 + 2sqrt{14} + 2)`
`= (9 - 2sqrt{14}) + (9 + 2sqrt{14})`
`= 9 - 2sqrt{14} + 9 + 2sqrt{14}`
`= (9 + 9) + (- 2sqrt{14} + 2sqrt{14})`
`= 18`
j) `{2}/{sqrt{3} - 1} - {1}/{sqrt{3} + 2}`
`= {2cdot(sqrt{3} + 1)}/{(sqrt{3} - 1)cdot(sqrt{3} + 1)} - {sqrt{3} - 2}/{(sqrt{3} + 2)cdot(sqrt{3} - 2)}`
`= {2cdot(sqrt{3} + 1)}/{(sqrt{3})^2 - 1^2} - {sqrt{3} - 2}/{(sqrt{3})^2 - 2^2}`
`= {2cdot(sqrt{3} + 1)}/{3 - 1} - {sqrt{3} - 2}/{3 - 4}`
`= {2cdot(sqrt{3} + 1)}/{2} - {sqrt{3} - 2}/{-1}`
`= (sqrt{3} + 1) + (sqrt{3} - 2)`
`= sqrt{3} + 1 + sqrt{3} - 2`
`= (sqrt{3} + sqrt{3}) + (1 - 2)`
`= 2sqrt{3} - 1`
k) `sqrt{4 - sqrt{7}} - sqrt{4 + sqrt{7}}`
Đặt `K = sqrt{4 - sqrt{7}} - sqrt{4 + sqrt{7}}`
`↔ sqrt{2}K = sqrt{2}cdot(sqrt{4 - sqrt{7}} - sqrt{4 + sqrt{7}})`
`= sqrt{2}cdot sqrt{4 - sqrt{7}} - sqrt{2}cdot sqrt{4 + sqrt{7}}`
`= sqrt{2cdot(4 - sqrt{7})} - sqrt{2cdot(4 + sqrt{7})}`
`= sqrt{8 - 2sqrt{7}} - sqrt{8 + 2sqrt{7}}`
`= sqrt{7 - 2sqrt{7} + 1} - sqrt{7 + 2sqrt{7} + 1}`
`= sqrt{(sqrt{7})^2 - 2cdot sqrt{7}cdot1 + 1^2} - sqrt{(sqrt{7})^2 + 2cdot sqrt{7}cdot1 + 1^2}`
`= sqrt{(sqrt{7} - 1)^2} - sqrt{(sqrt{7} + 1)^2}`
`= |sqrt{7} - 1| - |sqrt{7} + 1|`
`= (sqrt{7} - 1) - (sqrt{7} + 1)` (Vì `sqrt{7} > 1`)
`= sqrt{7} - 1 - sqrt{7} - 1`
`= (sqrt{7} - sqrt{7}) + (- 1 - 1)`
`= -2`
`→ K = -2 ÷ sqrt{2} = - sqrt{2}`
