Should it not be 보이어요(-다 + 어요)?
I read in TTMK(talktomeinkorean) that you conjugate the verb in present tense by adding different suffix after you remove 다.

Though the verb below seem to be a little bit of an exception?
Can you help me ^_^?

보이다 [bo-i-da] = to be seen, to be visible
Verb stem? 보이 [bo-i] What is it followed by? 어요 [eo-yo]

보이 + 어요 --- 보여요 [bo-yeo-yo] 보이다 [bo-i-da] = to be seen, to be visible
보여요 [bo-yeo-yo] = It’s visible. I see it.

But rather than보여요 should it not be 보이어요(-다 + 어요)? Where does the 여 comes from?
http://www.talktomeinkorean.com/lessons/l1l16 You can find the information in the pdf.

Question

Should it not be 보이어요(-다 + 어요)?
I read in TTMK(talktomeinkorean) that you conjugate the verb in present tense by adding different suffix after you remove 다.

Though the verb below seem to be a little bit of an exception? 
Can you help me ^_^?

보이다 [bo-i-da] = to be seen, to be visible
Verb stem? 보이 [bo-i] What is it followed by? 어요 [eo-yo]

보이 + 어요 ---> 보여요 [bo-yeo-yo] 보이다 [bo-i-da] = to be seen, to be visible
보여요 [bo-yeo-yo] = It’s visible. I see it.

But rather than보여요 should it not be 보이어요(-다 + 어요)? Where does the 여 comes from? 
http://www.talktomeinkorean.com/lessons/l1l16 You can find the information in the pdf.

Answer

Try saying 이어 fast ... I believe what happens is that you'll get with a Korean toung something that fairly well resembels 여 so to save time writing in these cases it somewhat strays from the very basic rule for any other verb ending on the vowel "i". For example:
기다리다 -> 잠깐만 기다려요 = Hold one a sec...(hope thats right)

Just be aware that Korean has several forms of politness which change the ending ..and also there are "irrigular" verbs which are especially verbs ending on 으 ㄹ ㄷ and maby others I don't know

Just be aware

Answer

Mojave, thank you very much.

Yeah, I get your point. :D ^_^ I get it now.

Answer

It's not an exception. Vowels regularly join and have a different shape. 보이다 -> 보이 + 어요 = 보여요. 보다 -> 보 + 아요 = 봐요. 하다 -> 하 + 여요 = 해요. Those are all listed in that that lesson you referenced.