\(\dfrac{\sqrt{5}}{\sqrt{3} + 1}\:-\:\sqrt{\dfrac{30}{8}} + \dfrac{\sqrt{45}}{2} = \dotso\)
Jawaban: B
\(\dfrac{\sqrt{5}}{\sqrt{3} + 1}\times \color{red} \dfrac{\sqrt{3} \:-\: 1}{\sqrt{3} \:-\: 1} \color{black}\:-\:\sqrt{\dfrac{15}{4}} + \dfrac{\sqrt{45}}{2}\)
\(\dfrac{\sqrt{5}(\sqrt{3} \:-\: 1)}{3\:-\:1}\:-\:\dfrac{\sqrt{15}}{2}+ \dfrac{\sqrt{9 \times 5}}{2}\)
\(\dfrac{\sqrt{15}\:-\:\sqrt{5})}{2}\:-\:\dfrac{\sqrt{15}}{2}+ \dfrac{3\sqrt{5}}{2}\)
\(\dfrac{\sqrt{15}}{2}\:-\:\dfrac{\sqrt{5}}{2}\:-\:\dfrac{\sqrt{15}}{2}+ \dfrac{3\sqrt{5}}{2}\)
\(\cancel{\dfrac{\sqrt{15}}{2}}\:-\:\dfrac{\sqrt{5}}{2}\:-\:\cancel{\dfrac{\sqrt{15}}{2}}+ \dfrac{3\sqrt{5}}{2}\)
\(-\dfrac{\sqrt{5}}{2} + \dfrac{3\sqrt{5}}{2}\)
\(\dfrac{\cancel{2}\sqrt{5}}{\cancel{2}}\)
\(\color{red} \sqrt{5}\)
Jawaban: D
\(\text{Misalkan } x = \sqrt{12 + \sqrt{12 + \sqrt{12 + \dotso }}}\:\:\:\:\:\color{blue}\text{kuadratkan kedua ruas}\)
\(x^2 = 12 + \sqrt{12 + \sqrt{12 + \dotso }}\)
\(x^2 = 12 + x\)
\(x^2 – x – 12 = 0\:\:\:\:\:\color{blue}\text{faktorkan}\)
\((x – 4)(x + 3) = 0\)
\(x – 4 = 0 \rightarrow x = 4\)
\(x + 3 = 0 \rightarrow x = -3\:\:\:\:\:\color{blue}\text{tidak memenuhi}\)
Karena hasil akar pangkat dua positif, maka ambil nilai \(x = 4\)
Jadi, \(\sqrt{12 + \sqrt{12 + \sqrt{12 + \dotso }}}= 4\)
Jawaban: D
Kerjakan dahulu \((-\sqrt{2} + \sqrt{3} + 2 – \sqrt{5})(\sqrt{2} + \sqrt{3} + 2 + \sqrt{5})\) dengan mengatur angkanya:
\([(2 + \sqrt{3}) – (\sqrt{2} + \sqrt{5})][(2 + \sqrt{3}) + (\sqrt{2} + \sqrt{5})]\)
\(\color{blue}\text{Note: } (a – b)(a + b) = a^2 – b^2\)
\([(2 + \sqrt{3}) – (\sqrt{2} + \sqrt{5})][(2 + \sqrt{3}) + (\sqrt{2} + \sqrt{5})] = (2 + \sqrt{3})^2 – (\sqrt{2} + \sqrt{5})^2\)
\(4 + 4\sqrt{3} + 3 – (2 + 2\sqrt{10} + 5)\)
\(7 + 4\sqrt{3} – (7 + 2\sqrt{10})\)
\(\cancel{7} + 4\sqrt{3} – \cancel{7} – 2\sqrt{10}\)
\(4\sqrt{3} – 2\sqrt{10}\)
Sekarang kita kerjakan \((4\sqrt{3} – 2\sqrt{10})(4\sqrt{3} + 2\sqrt{10})\)
\((4\sqrt{3})^2 – (2\sqrt{10})^2\)
\(4^2\cdot 3 – 2^2 \cdot 10\)
\(48 – 40\)
\(8\)
Bentuk sederhana dari \(\sqrt{10 + 2(\sqrt{15} + \sqrt{10} + \sqrt{6})}\) adalah ….
Jawaban: D
\(\sqrt{10 + 2(\sqrt{15} + \sqrt{10} + \sqrt{6})}\)
\(\sqrt{10 + 2\sqrt{15} + 2\sqrt{10} + 2\sqrt{6}}\)
\(\sqrt{2(5 + \sqrt{15}) + 2(\sqrt{10} + \sqrt{6})}\)
\(\sqrt{2(5 + \sqrt{15}) + 2\sqrt{16 + 2\sqrt{60}}}\)
\(\sqrt{2(5 + \sqrt{15}) + 2\sqrt{16 + 4\sqrt{15}}}\)
\(\sqrt{2(5 + \sqrt{15}) + 4\sqrt{4 + \sqrt{15}}}\)
\(\sqrt{2(5 + \sqrt{15} + 2\sqrt{4 + \sqrt{15}})}\)
\(\sqrt{2}\sqrt{(5 + \sqrt{15} + 2\sqrt{4 + \sqrt{15}})}\)
\(\sqrt{2}(\sqrt{4 + \sqrt{15}} + 1)\)
\(\color{blue}\sqrt{4 + \sqrt{15}} = \sqrt{\dfrac{8 + 2\sqrt{15}}{2}}\)
\(\color{blue}\sqrt{4 + \sqrt{15}} = \dfrac{\sqrt{5} + \sqrt{3}}{\sqrt{2}}\)
\(\sqrt{2}( \dfrac{\sqrt{5} + \sqrt{3}}{\sqrt{2}} + 1)\)
\(\color{purple} \sqrt{5} + \sqrt{3} + \sqrt{2}\)
Hasil dari \(\sqrt[3] {5 + 2\sqrt{13}} \:-\: \sqrt[3] {2\sqrt{13} \:-\:5}\) adalah …
Jawaban: A
\(\sqrt[3] {5 + 2\sqrt{13}} \:-\: \sqrt[3] {2\sqrt{13} \:-\:5} = p\)
Kedua ruas dipangkatkan 3
\(\left(\sqrt[3] {5 + 2\sqrt{13}} \:-\: \sqrt[3] {2\sqrt{13} \:-\:5}\right)^3 = p^3\)
Note:
\(\color{red} (a\:-\:b)^3 = a^3\:-\:3a^2b + 3ab^2\:-\:b^3\)
\(5 + \cancel{2\sqrt{13}}\:-\:3\left(\sqrt[3] {5 + 2\sqrt{13}}\right)^2 \sqrt[3] {2\sqrt{13} \:-\:5} + 3\sqrt[3] {5 + 2\sqrt{13}} \left(\sqrt[3] {2\sqrt{13} \:-\:5}\right)^2 \:-\: (\cancel{2\sqrt{13}} \:-\:5) = p^3\)
\(10\:-\:3\sqrt[3] {5 + 2\sqrt{13}}\cdot\sqrt[3] {(2\sqrt{13})^2\:-\:5^2} + 3\sqrt[3] {2\sqrt{13} \:-\:5}\cdot\sqrt[3] {(2\sqrt{13})^2 \:-\:5^2} = p^3\)
\(10\:-\:3\sqrt[3] {5 + 2\sqrt{13}}\cdot\sqrt[3] {52\:-\:25} + 3\sqrt[3] {2\sqrt{13} \:-\:5}\cdot\sqrt[3] {52\:-\:25} = p^3\)
\(10\:-\:3\sqrt[3] {5 + 2\sqrt{13}}\cdot\sqrt[3] {27} + 3\sqrt[3] {2\sqrt{13} \:-\:5}\cdot\sqrt[3] {27} = p^3\)
\(10\:-\:9\sqrt[3] {5 + 2\sqrt{13}}+ 9\sqrt[3] {2\sqrt{13} \:-\:5} = p^3\)
\(10\:-\:9\left(\sqrt[3] {5 + 2\sqrt{13}}\:-\: \sqrt[3] {2\sqrt{13}\:-\:5}\right) = p^3\)
\(10\:-\:9p = p^3\)
\(p^3 + 9p \:-\:10 = 0\)
\(p = 1\)
\(\text{Jadi}, \sqrt[3] {5 + 2\sqrt{13}} \:-\: \sqrt[3] {2\sqrt{13} \:-\:5} = 1\)
