<s>[INST] Problem based on Algebraic identities Algebraic identities:
$$(a+b)^2 = a^2 + b^2 + 2ab$$
$$(a-b)^2 = a^2 + b^2 - 2ab$$
Other identities can also come to solve this question?
If $x + 1/x = 5$ and $x^2 + 1/x^3 = 8$, then what would be the value of $x^3 + 1/x^2$?
Possible answers:
a- 215
b- 125
c- 256
d- 525
 [/INST] let $$a=x^2+\dfrac{1}{x^3},b=x^3+\dfrac{1}{x^2}$$
then $$a+b=x^2+\dfrac{1}{x^2}+x^3+\dfrac{1}{x^3}$$
and
$$x^2+\dfrac{1}{x^2}=(x+\dfrac{1}{x})^2-2=23$$
$$x^3+\dfrac{1}{x^3}=(x+\dfrac{1}{x})^3-3(x+\dfrac{1}{x})=125-15=110$$
so
$$a+b=133$$
then
$$b=133-8=125$$
</s>