Astronomy - Distance with different scale factors

[tosv]

Homework Statement
Expansion of the universe is described by the scale factor R(t), where t is the time after the Big Bang. For a flat universe the scale factor is today
R(t)=C_{1}\cdot t^{\frac{2}{3}}

When the Universe was radiation dominated, for t <200,000 years, the scale factor was
R(t)=C_{2}\cdot \sqrt{t}

Today, about 12 billion years after the Big Bang, we measure the distance to another superhop to 1 billion light years. How big would the distance between these sites one year after the Big Bang, if we assume that we live in a flat universe.

The attempt at a solution
I express the distance in meters
R(t)=9.461\cdot 10^{24}\, m

Then I express the time in seconds
t=3.787\cdot 10^{17}\, s

I found that the constant is
C_{1}=1.8075\cdot 10^{13}

I need to know the constant C_{2}, but I'm not sure how I'll be able to calculate it.

I have tested just to put C_{1}=C_{2}, but the result for the distance was wrong. I then realized that the two constants have different units.

Does anyone have a suggestion how I may proceed?

 

[Redbelly98]

For t = 200,000 years, both expressions for R(t) should give the same result.

To make your life easier, consider using units of years and light-years rather than seconds and meters.

 

[tosv]

Thanks for your reply, now I was able to solve the problem.

Thanks for your advice, I'll keep it in mind.

 

[Redbelly98]

Glad it worked out.

I know we get SI units drilled into our heads in intro physics classes, but they are not always the most convenient. If the problem statement uses alternate units, that's a hint for you to consider it too.