Question

Calcule: (1,222..)²-(1/3)²=?

Answer 1

Primeiro vamos transformar a dízima periódica $1,222\ldots$ em sua fração geratriz:

$x=1,222\ldots\\ \\ 10x=12,222\ldots\\ \\ \\ 10x-x=12,\mathbf{222\ldots}-1,\mathbf{222\ldots}\\ \\ 9x=11+1,\mathbf{222\ldots}-1,\mathbf{222\ldots}\\ \\ 9x=11\\ \\ x=\dfrac{11}{9}\\ \\ 1,222\ldots=\dfrac{11}{9}$


Então

$\left(1,222\ldots \right )^{2}-\left(\dfrac{1}{3} \right )^{2}\\ \\ =\left(\dfrac{11}{9} \right )^{2}-\left(\dfrac{1}{3} \right )^{2}\\ \\ =\dfrac{11^{2}}{9^{2}}-\dfrac{1^{2}}{3^{2}}\\ \\ =\dfrac{121}{81}-\dfrac{1}{9}\\ \\ =\dfrac{121}{81}-\dfrac{9}{81}\\ \\ =\dfrac{121-9}{81}\\ \\ =\dfrac{112}{81}$