Calculating age of the universe using redshift?

[zeion]

Hello. This is one of my coursework questions I was wondering if I could get some insight here.. here is the question:

 

The size of the Universe if conveniently parameterized by a scale factor, a(t), which simply describes how big the Universe is at other times relative to its present size (ie. at the present we say that a is 1, and at some time in the past when the Universe was half as big as it was today, then a was 0.5). A matter-dominated Universe grows with time as [math]a \propto t^{\frac{2}{3}} [/math]. Assuming the Universe is 13.5 billion years old at present, how old is the Universe at redshifts, z, of z = 0.5 ... etc, z= 100? Assume that we presently live in a matter-dominated Universe, and that the Universe is matter-dominated out to redshifts of at least 100.

 

 

The formula for redshift relative to scale factor is [math] 1 + z = \frac{a_{now}}{a_{then}} [/math] Then, since [math]a \propto t^{\frac{2}{3}}[/math] then [math] 1 + z = t^{\frac{2}{3}}[/math]

Then I plug in z and solve for t, then divide the current age by t?