« Reply #2 on: Sunday 07 April 13 15:46 BST (UK) »
the 1841 census shows Rachael as 4, so she was born before the start of civil registration, the most you will find is a baptism, and that may, or may not, show a date of birth, and may just show the abode as Trevethin.
1841 Cambrian Row, Hamlet of Cwmdu, Glam HO107/1422/16 folio 9 pg 10
William Adams 40 miner
Rachael 35
Ann 14
Thomas 13
Elizabeth 9
Solomon 7
Samuel 5
Rachael 4
Sarah 2
Catherine 4 months
only Catherine born in county, all the others born outside of Glamorgan
I've not found a baptism in the established church for Rachel, but there is a non-conformist one for Solomon
https://familysearch.org/pal:/MM9.1.1/FQP8-RPJthe 636 in the entry ref identifies the baptism as being in the Wesleyan Chapel in Trevethin, the survival of non-conformist records is patchy and the ones that do survive may not have been deposited at record offices.
Cornwall: Allen, Bevan, Bosisto, Carnpezzack, Donithorn, Huddy, James, Retallack, Russell, Vincent, Yeoman
Cards: Thomas (Llanbadarn Fawr)
Glam: Bowler, Cram, Galloway, James, Thomas, Watkins
Lincs: Coupland, Cram
Mon: Cram, Gwyn, John, Philpot, Smart, Watkins
Pembs: Edwards (St. Dogmael's)
Yorks: Airey, Bowler, Elliott, Hare, Hewitt, Kellett, Kemp, Stephenson, Tebb