Evaluate the following:

Question:

Evaluate the following:

(i) $(\sqrt{x+1}+\sqrt{x-1})^{6}+(\sqrt{x+1}-\sqrt{x-1})^{6}$

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

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

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

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

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

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

(viii) $(0.99)^{5}+(1.01)^{5}$

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

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

Solution:

(i) $(\sqrt{x+1}+\sqrt{x-1})^{6}+(\sqrt{x+1}-\sqrt{x-1})^{6}$

$=2\left[{ }^{6} C_{0}(\sqrt{x+1})^{6}(\sqrt{x-1})^{0}+{ }^{6} C_{2}(\sqrt{x+1})^{4}(\sqrt{x-1})^{2}+{ }^{6} C_{4}(\sqrt{x+1})^{2}(\sqrt{x-1})^{4}+{ }^{6} C_{6}(\sqrt{x+1})^{0}(\sqrt{x-1})^{6}\right]$

$=2\left[(x+1)^{3}+15(x+1)^{2}(x-1)+15(x+1)(x-1)^{2}+(x-1)^{3}\right.$

$=2\left[x^{3}+1+3 x+3 x^{2}+15\left(x^{2}+2 x+1\right)(x-1)+15(x+1)\left(x^{2}+1-2 x\right)+x^{3}-1+3 x-3 x^{2}\right]$

$=2\left[2 x^{3}+6 x+15 x^{3}-15 x^{2}+30 x^{2}-30 x+15 x-15+15 x^{3}+15 x^{2}-30 x^{2}-30 x+15 x+15\right]$

$=2\left[32 x^{3}-24 x\right]$

$=16 x\left[4 x^{2}-3\right]$

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

$=2\left[{ }^{6} C_{0} x^{6}\left(\sqrt{x^{2}-1}\right)^{0}+{ }^{6} C_{2} x^{4}\left(\sqrt{x^{2}-1}\right)^{2}+{ }^{6} C_{4} x^{2}\left(\sqrt{x^{2}-1}\right)^{4}+{ }^{6} C_{6} x^{0}\left(\sqrt{x^{2}-1}\right)^{6}\right]$

$=2\left[x^{6}+15 x^{4}\left(x^{2}-1\right)+15 x^{2}\left(x^{2}-1\right)^{2}+\left(x^{2}-1\right)^{3}\right]$

$=2\left[x^{6}+15 x^{6}-15 x^{4}+15 x^{2}\left(x^{4}-2 x^{2}+1\right)+\left(x^{6}-1+3 x^{2}-3 x^{4}\right)\right]$

$=2\left[x^{6}+15 x^{6}-15 x^{4}+15 x^{6}-30 x^{4}+15 x^{2}+x^{6}-1+3 x^{2}-3 x^{4}\right]$

$=64 x^{6}-96 x^{4}+36 x^{2}-2$

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

$=2\left[{ }^{5} C_{0}(2 \sqrt{x})^{0}+{ }^{5} C_{2}(2 \sqrt{x})^{2}+{ }^{5} C_{4}(2 \sqrt{x})^{4}\right]$

$=2\left[1+10 \times 4 x+5 \times 16 x^{2}\right]$

$=2\left[1+40 x+80 x^{2}\right]$

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

$=2\left[{ }^{6} C_{0}(\sqrt{2})^{6}+{ }^{6} C_{2}(\sqrt{2})^{4}+{ }^{6} C_{4}(\sqrt{2})^{2}+{ }^{6} C_{6}(\sqrt{2})^{0}\right]$

$=2[8+15 \times 4+15 \times 2+1)$

$=2 \times 99=198$

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

$=2\left[{ }^{5} C_{1} \times 3^{4} \times(\sqrt{2})^{1}+{ }^{5} C_{3} \times 3^{2} \times(\sqrt{2})^{3}+{ }^{5} C_{5} \times 3^{0} \times(\sqrt{2})^{5}\right]$

$=2[5 \times 81 \times \sqrt{2}+10 \times 9 \times 2 \sqrt{2}+4 \sqrt{2}]$

$=2 \sqrt{2}(405+180+4)=1178 \sqrt{2}$

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

$=2\left[{ }^{7} C_{0} \times 2^{7} \times(\sqrt{3})^{0}+{ }^{7} C_{2} \times 2^{5} \times(\sqrt{3})^{2}+{ }^{7} C_{4} \times 2^{3} \times(\sqrt{3})^{4}+{ }^{7} C_{6} \times 2^{1} \times(\sqrt{3})^{6}\right]$

$=2[128+21 \times 32 \times 3+35 \times 8 \times 9+7 \times 2 \times 27]$

$=2[128+2016+2520+378]$

$=2 \times 5042=10084$

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

$=2\left[{ }^{5} C_{1} \times(\sqrt{3})^{4}+{ }^{5} C_{3} \times(\sqrt{3})^{2}+{ }^{5} C_{5} \times(\sqrt{3})^{0}\right]$

$=2[5 \times 9+10 \times 3+1]$

$=2 \times 76=152$

(viii) $(0.99)^{5}+(1.01)^{5}$

$=(1-0.01)^{5}+(1+0.01)^{5}$

$=2\left[{ }^{5} C_{0}(0.01)^{0}+{ }^{5} C_{2}(0.01)^{2}+{ }^{5} C_{4}(0.01)^{4}\right]$

$=2[1+10 \times 0.0001+5 \times 0.00000001]$

$=2 \times 1.00100005=2.0020001$

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

$=2\left[{ }^{6} C_{1}(\sqrt{3})^{5}(\sqrt{2})^{1}+{ }^{6} C_{3}(\sqrt{3})^{3}(\sqrt{2})^{3}+{ }^{6} C_{5}(\sqrt{3})^{1}(\sqrt{2})^{5}\right]$

$=2[6 \times 9 \sqrt{3} \times \sqrt{2}+20 \times 3 \sqrt{3} \times 2 \sqrt{2}+6 \times \sqrt{3} \times 4 \sqrt{2}]$

$=2[\sqrt{6}(54+120+24)]$

$=396 \sqrt{6}$

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

$=2\left[{ }^{4} C_{0}\left(a^{2}\right)^{4}\left(\sqrt{a^{2}-1}\right)^{0}+{ }^{4} C_{2}\left(a^{2}\right)^{2}\left(\sqrt{a^{2}-1}\right)^{2}+{ }^{4} C_{4}\left(a^{2}\right)^{0}\left(\sqrt{a^{2}-1}\right)^{4}\right]$

$=2\left[a^{8}+6 a^{4}\left(a^{2}-1\right)+\left(a^{2}-1\right)^{2}\right]$

$=2\left[a^{8}+6 a^{6}-6 a^{4}+a^{4}+1-2 a^{2}\right]$

$=2 a^{8}+12 a^{6}-10 a^{4}-4 a^{2}+2$

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