Explicație pas cu pas:
a)
[tex]5 - 2 \sqrt{6} = {( \sqrt{3} - \sqrt{2} )}^{2} > 0 \ \implies 0 < a \\ [/tex]
[tex]\sqrt{3} - \sqrt{2} < \frac{1}{3} \iff {( \sqrt{3} - \sqrt{2} )}^{2} < \frac{1}{9} < \frac{1}{5} = 0.2 \\ \implies a < 0.2[/tex]
[tex]\implies \bf 0 < a < 0.2[/tex]
b)
[tex]{a}^{5} = {( \sqrt{3} - \sqrt{2} )}^{10}[/tex]
[tex] \sqrt{30} - \sqrt{20} \approx 1[/tex]
[tex] {\Big(\sqrt{10} ( \sqrt{3} - \sqrt{2})\Big)}^{10} \approx {1}^{10} [/tex]
[tex]{\Big( \sqrt{3} - \sqrt{2}\Big)}^{10} \cdot {10}^{5} \approx 1[/tex]
[tex]\implies {a}^{5} \approx {10}^{ - 5} \cdot 1[/tex]
a⁵ are primele patru zecimale egale cu 0
q.e.d.
