Question

Desenvolva os produtos notáveis:
(a/b + b/a)2

Answer 1

Dada a propriedade do quadrado da soma:

$\mathsf{\left(x+y\right)^2=x^2+2xy+y^2}$

Logo:

$\mathsf{\left(\dfrac{a}{b}+\dfrac{b}{a}\right)^2}\\\\\\\mathsf{\left(\dfrac{a\cdot a}{a\cdot b}+\dfrac{b\cdot b}{b\cdot a}\right)^2}\\\\\\\mathsf{\left(\dfrac{a^2}{ab}+\dfrac{b^2}{ab}\right)^2}\\\\\\\mathsf{\left(\dfrac{a^2+b^2}{ab}\right)^2}\\\\\\\mathsf{\dfrac{\left(a^2+b^2\right)^2}{\left(ab\right)^2}}\\\\\\\mathsf{\dfrac{\left(a^2\right)^2+2a^2b^2+\left(b^2\right)^2}{a^2b^2}}\\\\\\\boxed{\mathsf{\dfrac{a^4+2a^2b^2+b^4}{a^2b^2}\:\:\:\:ou\:\:\:\:\dfrac{a^2}{b^2}+2+\dfrac{b^2}{a^2}}}}\: \: \checkmark$