Question

Evaluate $(a+b{)}^5-(a-b{)}^5$, hence find $(\sqrt{3}+\sqrt{5}{)}^5-(\sqrt{3}-\sqrt{5}{)}^5$

Answer 1

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

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

$\left[\right(a+b{)}^3+(a-b{)}^3]\left[\right(a+b{)}^3-(a-b{)}^3]$

$(2a^3+6a{b}^2)(2b^3+6a^{2}b)$

$40a^3{b}^3+12a^{5}b+12a{b}^5+36a^3{b}^3$

$10a^2{b}^2+3a^4+3b^4$

$(a+b{)}^5-(a-b{)}^5=10a^2{b}^2+3a^4+3b^4$

$\therefore (\sqrt{3}+\sqrt{5}{)}^5-(\sqrt{3}-\sqrt{5}{)}^5=10\times \left(\sqrt{3}{)}^2\right(\sqrt{5}{)}^2+3(\sqrt{3}{)}^4+3(\sqrt{5}{)}^4$

$=10\times 3\times 5+3\left(9\right)+3\left(25\right)$

$=252$

$(\sqrt{3}+\sqrt{5}{)}^5-(\sqrt{3}-\sqrt{5}{)}^5=252$