Question

(PUC - RJ ) o valor da raiz quadrada de 1,777... sobre a raiz quadrada de 0,111. ..

Answer 1

Sistema 1:
x = 1,777...
10x = 17,777... (-)
9x = 16
x = 16/9

Sistema 2:
y = 0,111...
10y = 1,111... (-)
9y = 1
y = 1/9

√(16/9)/√(1/9) = (4/3)/(1/3) = 4/3 * 3 = 4 (corta-se os 3's)

Abraços

Answer 2

$\frac{ \sqrt{1,777...} }{ \sqrt{0,111...} } = \frac{ \sqrt{\frac{(17-1)}{9}} }{ \sqrt{\frac{1}{9}} } =\frac{ \sqrt{\frac{16}{9}} }{ \sqrt{\frac{1}{9}} } = \frac{4/3}{1/3} = \frac{4}{3} \cdot \frac{3}{1} = \frac{12}{3} = 4$