Question

Evaluate $(a+b{)}^5-(a-b{)}^5$ Hence find the value of $(\sqrt{3}+\sqrt{2}{)}^5-(\sqrt{3}-\sqrt{2}{)}^5$

Answer 1

$(a+b{)}^5-(a-b{)}^5$

$(a+b{)}^2(a+b{)}^2(a+b)$

$\Rightarrow (a^2+b^2+2ab)(a^2+b^2+2ab)(a+b)$

$\Rightarrow (a^4+a^2{b}^2+2a^{3}b+a^2{b}^2+b^4+2a{b}^3+2a^{3}b+$

$2a^2{b}^2+2a{b}^3\left)\right(a+b)$

$\Rightarrow [a^4+b^4+4a^2{b}^2+4a^{3}b+4a{b}^3](a+b)$

$\Rightarrow [a^5+a{b}^4+4a^3{b}^2+4a^{4}b+4a^2{b}^3+b{a}^4$

$+b^5+4a^2{b}^3+4a^3{b}^2+4a{b}^4]$

$\Rightarrow [a^5+b^5+8a^2{b}^3+8a^3{b}^2+$

$5a{b}^4+5a^{4}b]$

$\Rightarrow$ so $(a-b{)}^5$

$=a^5-b^5-8a^2{b}^3+8a^3{b}^2+5a{b}^4-5a^{4}b$

$(a+b{)}^5-(a-b{)}^5=a^5+b^5+8a^2{b}^3+8a^3{b}^2+$

$5a{b}^4+5a^{4}b-a^5+b^5+8a^2{b}^3-8a^3{b}^2$

$-5a{b}^4+5a^{4}b$

$\Rightarrow 2b^5+16a^2{b}^3+10a^{4}b$

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

$\Rightarrow 2(\sqrt{2}{)}^5+16(\sqrt{3}{)}^2(\sqrt{2}{)}^3+10(\sqrt{3}{)}^4\left(\sqrt{2}\right)$

$\Rightarrow 8\sqrt{2}+48\times 2\sqrt{2}+10\times 9\left(\sqrt{2}\right)$

$\Rightarrow 8\sqrt{2}+96\sqrt{2}+90\sqrt{2}$

$\Rightarrow 8\sqrt{2}+186\sqrt{2}\Rightarrow 194\sqrt{2}$