a) \( a = 2\sqrt{2}(4 - \sqrt{2}) \)
\( = 2\sqrt{2} \cdot 4 - 2\sqrt{2} \cdot \sqrt{2} \)
\( = 8\sqrt{2} - 2 \cdot 2 \)
\( = 8\sqrt{2} - 4 \)
b) \( b = \sqrt{3}(\sqrt{3} - 1) \)
\( = \sqrt{3} \cdot \sqrt{3} - \sqrt{3} \)
\( = 3 - \sqrt{3} \)
c) \( c = 5(\sqrt{2} - 2) \)
\( = 5\sqrt{2} - 10 \)
d) \( d = -2(5\sqrt{3} - \sqrt{2}) \)
\( = -10\sqrt{3} + 2\sqrt{2} \)
e) \( e = 3(2\sqrt{5} - 4) \)
\( = 6\sqrt{5} - 12 \)
f) \( f = \sqrt{2}(4\sqrt{2} - 3) - \sqrt{3}(2\sqrt{3} + 1) \)
\( = 4 \cdot 2 - 3\sqrt{2} - 2 \cdot 3 - \sqrt{3} \)
\( = 8 - 3\sqrt{2} - 6 - \sqrt{3} \)
\( = 2 - 3\sqrt{2} - \sqrt{3} \)
g) \( g = 2\sqrt{3}(4\sqrt{3} - 1) - 4(\sqrt{3} + 2) + 3\sqrt{3}(1 - \sqrt{3}) \)
\( = 2 \cdot 4 \cdot 3 - 2\sqrt{3} - 4\sqrt{3} - 8 + 3\sqrt{3} - 9 \)
\( = 24 - 3\sqrt{3} - 17 \)
\( = 7 - 3\sqrt{3} \)
h) \( h = 3(2\sqrt{2} - 3) - \sqrt{2}(4 - \sqrt{2}) + 5\sqrt{3}(4 - \sqrt{3}) \)
\( = 6\sqrt{2} - 9 - 4\sqrt{2} + 2 + 20\sqrt{3} - 5 \cdot 3 \)
\( = 2\sqrt{2} - 7 + 20\sqrt{3} - 15 \)
\( = 2\sqrt{2} + 20\sqrt{3} - 22 \)
i) \( i = 2(\sqrt{5} - 1) - 3(4\sqrt{5} - 3) \)
\( = 2\sqrt{5} - 2 - 12\sqrt{5} + 9 \)
\( = -10\sqrt{5} + 7 \)
k) \( k = (\sqrt{3} - 1)(2 - \sqrt{3}) \)
\( = 2\sqrt{3} - \sqrt{3}^2 - 2 + \sqrt{3} \)
\( = 2\sqrt{3} - 3 - 2 + \sqrt{3} \)
\( = 3\sqrt{3} - 5 \)
l) \( l = (\sqrt{2} - 2)(\sqrt{2} + 3) \)
\( = \sqrt{2}^2 + 3\sqrt{2} - 2\sqrt{2} - 6 \)
\( = 2 + \sqrt{2} - 6 \)
\( = \sqrt{2} - 4 \)
m) \( m = (2\sqrt{3} - 1)(3\sqrt{3} + 4) \)
\( = 6 \cdot 3 + 8\sqrt{3} - 3\sqrt{3} - 4 \)
\( = 18 + 5\sqrt{3} - 4 \)
\( = 14 + 5\sqrt{3} \)
n) \( n = (4 + \sqrt{3})(2\sqrt{3} - 1) + (3\sqrt{2} - 4)(\sqrt{2} - 1) \)
\( = 8\sqrt{3} - 4 + 2 \cdot 3 - \sqrt{3} + 3 \cdot 2 - 3\sqrt{2} - 4\sqrt{2} + 4 \)
\( = 8\sqrt{3} - 4 + 6 - \sqrt{3} + 6 - 3\sqrt{2} - 4\sqrt{2} + 4 \)
\( = 7\sqrt{3} + 12 - 7\sqrt{2} \)
p) \( p = (2\sqrt{5} + 2)(3 - 2\sqrt{5}) - (3\sqrt{3} - 2)(3 - 4\sqrt{3}) \)
\( = 6\sqrt{5} + 6 - 4 \cdot 5 - 4 - 9\sqrt{3} + 12 \cdot 3 + 6 - 8\sqrt{3} \)
\( = 6\sqrt{5} + 6 - 20 - 4 - 9\sqrt{3} + 36 + 6 - 8\sqrt{3} \)
\( = 6\sqrt{5} + 24 - 17\sqrt{3} \)