Question

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Answer 1

$\sqrt{ \frac{5}{0,0(2)} } + \sqrt{ \frac{55}{0,0(02) } } + \sqrt{ \frac{555}{ 0,0(002) } } + \sqrt{ \frac{5555}{ 0,0(0002) } } = ֍\\ \\ 0,0(2) = \frac{2}{(9)0} = \frac{1}{45} \\ \\ 0,0(02) = \frac{2}{(99)0} = \frac{1}{495} \\ \\ 0,0(002) = \frac{2}{9990} = \frac{1}{4995} \\ \\ 0,0(0002) = \frac{1}{49995} \\ \\ ֍ = \sqrt{5 \times 45} + \sqrt{55 \times 495} + \sqrt{555 \times 4995} + \sqrt{5555 \times 49995} = \\ \sqrt{25 \times 9} + \sqrt{25 \times 11 \times 99} + \sqrt{25 \times 111 \times 999} + \sqrt{25 \times 1111 \times 9999} = \\ \sqrt{ {5}^{2} \times {3}^{2} } + \sqrt{ {5}^{2} \times {11}^{2} \times {3}^{2} } + \sqrt{ {5}^{2} \times {111}^{2} \times {3}^{2} } + \sqrt{ {5}^{2} \times {1111}^{2} \times {3}^{2} } = \\ 5 \times 3 + 5 \times 11 \times 3 + 5 \times 111 \times 3 + 15 \times 1111∈ℕ$