Pomocy!!! Muszę zaliczyć semestr, jutro mam z tego sprawdzian i nic nie kumam.

1. a) [latex] frac{ sqrt{ sqrt{5} - 1 } * sqrt{ 1 + sqrt{5} } }{ 5 } = [/latex] [latex] = frac{ sqrt{ (sqrt{5} - 1)(1 + sqrt{5}) }}{ 5 } = [/latex] [latex] = frac{ sqrt{ (sqrt{5} - 1)(sqrt{5} + 1) }}{ 5 } = [/latex] [latex] = frac{ sqrt{ (sqrt{5})^2 - 1^2 }}{ 5 } = [/latex] [latex] = frac{ sqrt{ 5 - 1 }}{ 5 } = [/latex] [latex] = frac{ sqrt{4} }{ 5 } = [/latex] [latex] = frac{2}{5} [/latex] b) [latex] frac{ 5 + 2sqrt{3} }{ 3sqrt{3} - 12 } + frac{ sqrt{3} }{ 3 } = [/latex] [latex] = frac{ 5 + 2sqrt{3} }{ 3(sqrt{3} - 4) } + frac{ sqrt{3} }{ 3 } = [/latex] [latex] = frac{ 5 + 2sqrt{3} }{ 3(sqrt{3} - 4) } + frac{ sqrt{3}(sqrt{3} - 4) }{ 3(sqrt{3} - 4) } = [/latex] [latex] = frac{ 5 + 2sqrt{3} + sqrt{3}(sqrt{3} - 4) }{ 3(sqrt{3} - 4) } = [/latex] [latex] = frac{ 5 + 2sqrt{3} + 3 - 4sqrt{3} }{ 3(sqrt{3} - 4) } = [/latex] [latex] = frac{ 8 - 2sqrt{3} }{ 3(sqrt{3} - 4) } = [/latex] [latex] = frac{ 2(4 - sqrt{3}) }{ 3(sqrt{3} - 4) } = [/latex] [latex] = frac{ -2(sqrt{3} - 4) }{ 3(sqrt{3} - 4) } = [/latex] [latex] = frac{ -2 }{ 3 } = [/latex] [latex] = -frac{2}{3} [/latex] 2. a) [latex] a = sqrt{2} [/latex], [latex] b = sqrt{3} [/latex] [latex] (a + 2b)(2b - a) + (a + 3b)^2 - b(3b + a) = [/latex] [latex] = (2b + a)(2b - a) + a^2 + 6ab + 9b^2 - 3b^2 - ab = [/latex] [latex] = 4b^2 - a^2 + a^2 + 6ab + 9b^2 - 3b^2 - ab = [/latex] [latex] = 5ab + 10b^2 = [/latex] [latex] = 5 * sqrt{2} * sqrt{3} + 10 * (sqrt{3})^2 = [/latex] [latex] = 5sqrt{6} + 10 * 3 = [/latex] [latex] = 30 + 5sqrt{6} [/latex] b) [latex] b = -frac{1}{2} [/latex] [latex] frac{ (a + b)^2 - (ab + 1)^2 }{ a^2 - 1 } = [/latex] [latex] = frac{ a^2 + 2ab + b^2 - (a^2b^2 + 2ab + 1) }{ a^2 - 1 } = [/latex] [latex] = frac{ a^2 + 2ab + b^2 - a^2b^2 - 2ab - 1 }{ a^2 - 1 } = [/latex] [latex] = frac{ a^2 + b^2 - a^2b^2 - 1 }{ a^2 - 1 } = [/latex] [latex] = frac{ a^2 - 1 - a^2b^2 + b^2 }{ a^2 - 1 } = [/latex] [latex] = frac{ (a^2 - 1) - b^2(a^2 - 1) }{ a^2 - 1 } = [/latex] [latex] = frac{ (1 - b^2)(a^2 - 1) }{ a^2 - 1 } = [/latex] [latex] = 1 - b^2 = [/latex] [latex] = 1 - (- frac{1}{2})^2 = [/latex] [latex] = 1 - frac{1}{4} = [/latex] [latex] = frac{3}{4} [/latex] 3. a) [latex] 2 * 5^7 + 3 * 5^6 + 3 * 5^5 = [/latex] [latex] = 2 * 5^2 * 5^5 + 3 * 5 * 5^5 + 3 * 5^5 = [/latex] [latex] = 5^5(2 * 5^2 + 3 * 5 + 3) = [/latex] [latex] = 5^5(2 * 25 + 15 + 3) = [/latex] [latex] = 5^5(50 + 15 + 3) = [/latex] [latex] = 5^5 * 68 = [/latex] [latex] = 5^5 * 34 * 2 [/latex] Skoro nasza liczba da się zapisać jako iloczyn liczb całkowitych, w tym dwójki, to na pewno jest to liczba parzysta. b) [latex] 3^{12} + 9^7 + 27^5 = [/latex] [latex] = 3^{12} + (3^2)^7 + (3^3)^5 = [/latex] [latex] = 3^{12} + 3^{14} + 3^{15} = [/latex] [latex] = 3^{12}(1 + 3^2 + 3^3) = [/latex] [latex] = 3^{12}(1 + 9 + 27) = [/latex] [latex] = 3^{12} * 37 [/latex] Zapisaliśmy naszą liczbę jako iloczyn liczb całkowitych, w tym liczby 37, więc jest ona podzielna przez m.in. 37 właśnie. Co należało pokazać.