Question

ajutor va rog nu ma pricep la module delok


ex 10​

Answer 1

Explicație pas cu pas:

a)

$\sqrt{3} < \sqrt{5} \iff \sqrt{3} - \sqrt{5} < 0 \implies \\ | \sqrt{3} - \sqrt{5} | = - (\sqrt{3} - \sqrt{5}) = \sqrt{5} - \sqrt{3}$

$\sqrt{3} + \sqrt{5} > 0 \implies | \sqrt{3} + \sqrt{5} | = \sqrt{3} + \sqrt{5}$

$3 - 2 \sqrt{5} = \sqrt{9} - \sqrt{20} < 0 \implies \\ |3 - 2 \sqrt{5} | = - (3 - 2 \sqrt{5}) = 2 \sqrt{5} - 3$

=>

$| \sqrt{3} - \sqrt{5} | + | \sqrt{3} + \sqrt{5} | - |3 - 2 \sqrt{5} | = \sqrt{5} - \sqrt{3} + \sqrt{3} + \sqrt{5} - (2 \sqrt{5} - 3) = \sqrt{5} - \sqrt{3} + \sqrt{3} + \sqrt{5} - 2 \sqrt{5} + 3 = 2 \sqrt{5} - 2 \sqrt{5} + 3 = 3$

b)

$2 - \sqrt{5} = \sqrt{4} - \sqrt{5} < 0 \implies \\ |2 - \sqrt{5} | = - (2 - \sqrt{5}) = \sqrt{5} - 2$

$\sqrt{3} + \sqrt{5} > 0 \implies | \sqrt{3} + \sqrt{5} | = \sqrt{3} + \sqrt{5}$

$1 - \sqrt{3} = \sqrt{1} - \sqrt{3} < 0 \implies \\ |1 - \sqrt{3} | = - (1 - \sqrt{3}) = \sqrt{3} - 1$

$\sqrt{3} - 3 = \sqrt{3} - \sqrt{9} < 0 \implies \\ |\sqrt{3} - 3| = - (\sqrt{3} - 3) = 3 - \sqrt{3}$

=>

$|2 - \sqrt{5} | - | \sqrt{3} + \sqrt{5} | + |1 - \sqrt{3} | - | \sqrt{3} - 3| = \sqrt{5} - 2 - (\sqrt{3} + \sqrt{5}) + \sqrt{3} - 1 - (3 - \sqrt{3}) = \sqrt{5} - 2 - \sqrt{3} - \sqrt{5} + \sqrt{3} - 1 - 3 + \sqrt{3} = \sqrt{3} - 6$