Question
Simplify:$\left(a+\frac{1}{a}\right)^3-\left(a-\frac{1}{a}\right)^3$
✓
Answer
$\left(a+\frac{1}{a}\right)^3-\left(a-\frac{1}{a}\right)^3 $
$ =(a) 3+\left(\frac{1}{a}\right)^3+3(a)\left(\frac{1}{a}\right)\left(a+\frac{1}{a}\right)-\left[(a)^3-\left(\frac{1}{a}\right)^3=-3(a)\left(\frac{1}{a}\right)\left(a-\frac{1}{a}\right)\right] $
$ =a^3+\frac{1}{a^3}+3\left(a+\frac{1}{a}\right)-\left[a^3-\frac{1}{a^3}-3\left(a-\frac{1}{a}\right)\right]$
$ =a^3+\frac{1}{a^3}+3 a+\frac{3}{a}-a^3+\frac{1}{a^3}+3 a-\frac{3}{a} $
$ =\frac{2}{a^3}+6 a$
$ =(a) 3+\left(\frac{1}{a}\right)^3+3(a)\left(\frac{1}{a}\right)\left(a+\frac{1}{a}\right)-\left[(a)^3-\left(\frac{1}{a}\right)^3=-3(a)\left(\frac{1}{a}\right)\left(a-\frac{1}{a}\right)\right] $
$ =a^3+\frac{1}{a^3}+3\left(a+\frac{1}{a}\right)-\left[a^3-\frac{1}{a^3}-3\left(a-\frac{1}{a}\right)\right]$
$ =a^3+\frac{1}{a^3}+3 a+\frac{3}{a}-a^3+\frac{1}{a^3}+3 a-\frac{3}{a} $
$ =\frac{2}{a^3}+6 a$
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