Question

Evaluate the following:
(i) ${\left(\sqrt{x+1}+\sqrt{x-1}\right)}^6+{\left(\sqrt{x+1}-\sqrt{x-1}\right)}^6$

(ii) ${\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6$

(iii) ${\left(1+2\sqrt{x}\right)}^5+{\left(1-2\sqrt{x}\right)}^5$

(iv) ${\left(\sqrt{2}+1\right)}^6+{\left(\sqrt{2}-1\right)}^6$

(v) ${\left(3+\sqrt{2}\right)}^5-{\left(3-\sqrt{2}\right)}^5$

(vi) ${\left(2+\sqrt{3}\right)}^7+{\left(2-\sqrt{3}\right)}^7$

(vii) ${\left(\sqrt{3}+1\right)}^5-{\left(\sqrt{3}-1\right)}^5$

(viii) ${\left(0.99\right)}^5+{\left(1.01\right)}^5$

(ix) ${\left(\sqrt{3}+\sqrt{2}\right)}^6-{\left(\sqrt{3}-\sqrt{2}\right)}^6$

(x) ${\left\{a^2+\sqrt{a^2-1}\right\}}^4+{\left\{a^2-\sqrt{a^2-1}\right\}}^4$

Answer 1

(i)
$$\begin{gathered} (\sqrt{x+1}+\sqrt{x-1}{)}^6+(\sqrt{x+1}-\sqrt{x-1}{)}^6 \\ =2[{}^{6}C_0(\sqrt{x+1}{)}^6(\sqrt{x-1}{)}^0+{}^{6}C_2(\sqrt{x+1}{)}^4(\sqrt{x-1}{)}^2+{}^{6}C_4(\sqrt{x+1}{)}^2(\sqrt{x-1}{)}^4+{}^{6}C_6(\sqrt{x+1}{)}^0\left(\sqrt{x-1}{)}^6\right] \\ =2\left[\right(x+1{)}^3+15(x+1{)}^2(x-1)+15(x+1\left)\right(x-1{)}^2+(x-1{)}^3 \\ =2[x^3+1+3x+3x^2+15(x^2+2x+1)(x-1)+15(x+1)(x^2+1-2x)+x^3-1+3x-3x^2] \\ =2[2x^3+6x+15x^3-15x^2+30x^2-30x+15x-15+15x^3+15x^2-30x^2-30x+15x+15] \\ =2[32x^3-24x] \\ =16x[4x^2-3] \end{gathered}$$

(ii)
$$\begin{gathered} (x+\sqrt{x^2-1}{)}^6+(x-\sqrt{x^2-1}{)}^6 \\ =2\left[{}^{6}C_0{x}^6\right(\sqrt{x^2-1}{)}^0+{}^{6}C_2{x}^4(\sqrt{x^2-1}{)}^2+{}^{6}C_4{x}^2(\sqrt{x^2-1}{)}^4+{}^{6}C_6{x}^0\left(\sqrt{x^2-1}{)}^6\right] \\ =2[x^6+15x^4(x^2-1)+15x^2(x^2-1{)}^2+(x^2-1{)}^3] \\ =2[x^6+15x^6-15x^4+15x^2(x^4-2x^2+1)+(x^6-1+3x^2-3x^4\left)\right] \\ =2[x^6+15x^6-15x^4+15x^6-30x^4+15x^2+x^6-1+3x^2-3x^4] \\ =64x^6-96x^4+36x^2-2 \end{gathered}$$

(iii)
$$\begin{gathered} (1+2\sqrt{x}{)}^5+(1-2\sqrt{x}{)}^5 \\ =2\left[{}^{5}C_0\right(2\sqrt{x}{)}^0+{}^{5}C_2(2\sqrt{x}{)}^2+{}^{5}C_4(2\sqrt{x}{)}^4] \\ =2[1+10\times 4x+5\times 16x^2] \\ =2[1+40x+80x^2] \end{gathered}$$

(iv)
$$\begin{gathered} (\sqrt{2}+1{)}^6+(\sqrt{2}-1{)}^6 \\ =2\left[{}^{6}C_0\right(\sqrt{2}{)}^6+{}^{6}C_2(\sqrt{2}{)}^4+{}^{6}C_4(\sqrt{2}{)}^2+{}^{6}C_6\left(\sqrt{2}{)}^0\right] \\ =2[8+15\times 4+15\times 2+1) \\ =2\times 99=198 \end{gathered}$$

(v)
$$\begin{gathered} (3+\sqrt{2}{)}^5-(3-\sqrt{2}{)}^5 \\ =2\left[{}^{5}C_1\times 3^4\times (\sqrt{2}{)}^1+{}^{5}C_3\times 3^2\times (\sqrt{2}{)}^3+{}^{5}C_5\times 3^0\times (\sqrt{2}{)}^5\right] \end{gathered}$$

$$\begin{gathered} =2[5\times 81\times \sqrt{2}+10\times 9\times 2\sqrt{2}+4\sqrt{2}] \\ =2\sqrt{2}(405+180+4)=1178\sqrt{2} \end{gathered}$$

(vi)
$$\begin{gathered} (2+\sqrt{3}{)}^7+(2-\sqrt{3}{)}^7 \\ =2[{}^{7}C_0\times 2^7\times (\sqrt{3}{)}^0+{}^{7}C_2\times 2^5\times (\sqrt{3}{)}^2+{}^{7}C_4\times 2^3\times (\sqrt{3}{)}^4+{}^{7}C_6\times 2^1\times \left(\sqrt{3}{)}^6\right] \\ =2[128+21\times 32\times 3+35\times 8\times 9+7\times 2\times 27] \\ =2[128+2016+2520+378] \\ =2\times 5042=10084 \end{gathered}$$

(vii)
$$\begin{gathered} (\sqrt{3}+1{)}^5-(\sqrt{3}-1{)}^5 \\ =2[{}^{5}C_1\times (\sqrt{3}{)}^4+{}^{5}C_3\times (\sqrt{3}{)}^2+{}^{5}C_5\times (\sqrt{3}{)}^0] \\ =2[5\times 9+10\times 3+1] \\ =2\times 76=152 \end{gathered}$$

(viii)
$$\begin{gathered} (0.99{)}^5+(1.01{)}^5 \\ =(1-0.01{)}^5+(1+0.01{)}^5 \\ =2\left[{}^{5}C_0\right(0.01{)}^0+{}^{5}C_2(0.01{)}^2+{}^{5}C_4(0.01{)}^4] \\ =2[1+10\times 0.0001+5\times 0.00000001] \\ =2\times 1.00100005=2.0020001 \end{gathered}$$

(ix)
$$\begin{gathered} (\sqrt{3}+\sqrt{2}{)}^6-(\sqrt{3}-\sqrt{2}{)}^6 \\ =2\left[{}^{6}C_1\right(\sqrt{3}{)}^5(\sqrt{2}{)}^1+{}^{6}C_3(\sqrt{3}{)}^3(\sqrt{2}{)}^3+{}^{6}C_5(\sqrt{3}{)}^1(\sqrt{2}{)}^5] \end{gathered}$$
$$\begin{gathered} =2[6\times 9\sqrt{3}\times \sqrt{2}+20\times 3\sqrt{3}\times 2\sqrt{2}+6\times \sqrt{3}\times 4\sqrt{2}] \\ =2\left[\sqrt{6}\right(54+120+24\left)\right] \\ =396\sqrt{6} \end{gathered}$$

(x)
$$\begin{gathered} {\left\{a^2+\sqrt{a^2-1}\right\}}^4+{\left\{a^2-\sqrt{a^2-1}\right\}}^4 \\ =2\left[{}^{4}C_0\right(a^2{)}^4(\sqrt{a^2-1}{)}^0+{}^{4}C_2(a^2{)}^2(\sqrt{a^2-1}{)}^2+{}^{4}C_4(a^2{)}^0\left(\sqrt{a^2-1}{)}^4\right] \\ =2[a^8+6a^4(a^2-1)+(a^2-1{)}^2] \\ =2[a^8+6a^6-6a^4+a^4+1-2a^2] \\ =2a^8+12a^6-10a^4-4a^2+2 \end{gathered}$$