有理數 & 無理數

1. √578 = √(2 * 17 * 17) = 17√2
2. √12 + √75 - √108
= 2√3 + 5√3 - 6√3
= √3
3. √125 * √80
= √10000
= 100
(4) (√3 - √2)(2√2 + √3)
= 2√6 - 4 + 3 - √6
= -1 + √6
(5) 2 / (2√3 - 1)
= 2(2√3 + 1) /[(2√3 - 1)(2√3 + 1)]
= (4√3 + 2) / (12 - 1)
= (4√3 + 2)/11
(6a) (√10 - 3)(√10 + 3)
= (√10)^2 - 3^2
= 10 - 9
= 1
(b) 看不清楚：
是(√10 - 3)^5(√10 + 3)^5
= 1^5 = 1
還是:
(√10 - 3)^5(√10 + 3)^3
= (√10 - 3)^2(1^3)
= 10 - 6√10 + 9
= 19 - 6√10
(7) P = √2 + √2 + 1 + √2 + 2 = 3 + 3√2
Q = √2 + √2 - 1 + √2 - 2 = 3√2 - 3
(a) P + Q = 6√2 無理
(b) P - Q = 6 有理
(c) PQ = (3 + 3√2)(3√2 - 3) = 18 - 9 = 9 無理
(8) a = 3 - √6
3/a = 3 /(3 - √6)
= 3(3 + √6 / (9 - 6)
= 3 + √6
a + 3/a = 3 - √6 + 3 + √6 = 6
(9) 5 /(√3 + √2) - 1 / (√3 - √2)
= [5(√3 - √2) - (√3 + √2)] / [(√3 + √2)(√3 - √2)]
= 4√3 - 6√2
(10)(a) (3√6 + 2√2)^2
= 54 + 12√12 + 8
= 62 + 24√3
(b) (3√6 + 2√2)(3√6 - 2√2)
= (3√6)^2 - (2√2)^2
= 54 - 8
= 46