Hướng dẫn trả lời:
Bài 1:
a) `{x^2 - 5}/{x + sqrt{5}}` (`x ne - sqrt{5}`)
`= {x^2 - (sqrt{5})^2}/{x + sqrt{5}}`
`= {(x + sqrt{5})cdot(x - sqrt{5})}/{x + sqrt{5}}`
`= x - sqrt{5}`
b) `{x^2 + 2sqrt{2}x + 2}/{x^2 - 2}` (`x ne ± sqrt{2}`)
`= {x^2 + 2cdotxcdot sqrt{2} + (sqrt{2})^2}/{x^2 - (sqrt{2})^2}`
`= {(x + sqrt{2})^2}/{(x + sqrt{2})cdot(x - sqrt{2})}`
`= {x + sqrt{2}}/{x - sqrt{2}}`
c) `{x - sqrt{xy}}/{x - y} + {sqrt{y}}/{sqrt{x} + sqrt{y}}`
`= {x - sqrt{xy}}/{x - y} + {sqrt{y}cdot(sqrt{x} - sqrt{y})}/{(sqrt{x} + sqrt{y})cdot(sqrt{x} - sqrt{y})}`
`= {x - sqrt{xy}}/{x - y} + {sqrt{y}cdot(sqrt{x} - sqrt{y})}/{(sqrt{x})^2 - (sqrt{y})^2}`
`= {x - sqrt{xy}}/{x - y} + {sqrt{y}cdot(sqrt{x} - sqrt{y})}/{x - y}`
`= {x - sqrt{xy} + sqrt{y}cdot(sqrt{x} - sqrt{y})}/{x - y}`
`= {x - sqrt{xy} + sqrt{y}cdot sqrt{x} + sqrt{y}cdot(- sqrt{y})}/{x - y}`
`= {x - sqrt{xy} + sqrt{xy} - y}/{x - y}`
`= {x - y}/{x - y}`
`= 1`
d) `{sqrt{a} - a}/{sqrt{a} - 1}`
`= {sqrt{a} - (sqrt{a})^2}/{sqrt{a} - 1}`
`= {sqrt{a}cdot(1 - sqrt{a})}/{sqrt{a} - 1}`
`= {-sqrt{a}cdot(sqrt{a} - 1)}/{sqrt{a} - 1}`
`= - sqrt{a}`
e) `{3 + sqrt{3}}/{sqrt{6} + sqrt{2}}`
`= {(sqrt{3})^2 + sqrt{3}}/{sqrt{2}cdot sqrt{3} + sqrt{2}}`
`= {sqrt{3}cdot(sqrt{3} + 1)}/{sqrt{2}cdot(sqrt{3} + 1)}`
`= sqrt{3}/sqrt{2}`
g) `{xsqrt{y} + ysqrt{x}}/{x + 2sqrt{xy} + y}`
`= {sqrt{xy}cdot sqrt{x} + sqrt{xy}cdot sqrt{y}}/{(sqrt{x})^2 + 2cdot sqrt{x}cdot sqrt{y} + (sqrt{y})^2}`
`= {sqrt{xy}cdot(sqrt{x} + sqrt{y})}/{(sqrt{x} + sqrt{y})^2}`
`= sqrt{xy}/{sqrt{x} + sqrt{y}}`
Đáp án:
a) `= x - sqrt{5}`
b) `= {x + sqrt{2}}/{x - sqrt{2}}`
c) `= 1`
d) `= - sqrt{a}`
e) `= sqrt{3}/sqrt{2}`
g) `= sqrt{xy}/{sqrt{x} + sqrt{y}}`