How can I prove the pattern $\sqrt{1 + 155555…5} = 2 \sqrt{3888…89}?$ How can I prove this 

$$\sqrt{1+155}=2\sqrt{39}$$
  $$\sqrt{1+1555}=2\sqrt{389}$$
  $$\sqrt{1+15555}=2\sqrt{3889}$$
  $$\sqrt{1+155555}=2\sqrt{38889}$$


The expressions on the right are 
$$
\sqrt{4\left( 3\cdot 10^{k-1} + \frac{8}{9}\cdot 10^{k-1} - \frac{80}{9} + 9\right) }$$
for $k \geq 2$.  I'm going to compare that to the expressions on the left, which have $k$ 5's. 
$$
\sqrt{4\left( 3\cdot 10^{k-1} + \frac{8}{9}\cdot 10^{k-1} - \frac{80}{9} + 9\right) }
= \sqrt{4\left( \frac{35}{9} \cdot 10^{k-1} + \frac{1}{9}\right)} = \sqrt{\frac{140}{9}\cdot 10^{k-1} + \frac{4}{9} } = \sqrt{\frac{14}{9} \cdot 10^k + 1 - \frac{5}{9} }= \sqrt{1 + 10^k + \frac{5}{9}\cdot 10^k - \frac{5}{9} }
$$
and the latter is the expression on the left.
