Te Form + oku, okimasu
~て+おく oku: It means "will do certainly", "do in advance".
~て+おきます okimasu same but polite.
By itself, おく oku means "to put," but after a verb in the Te Form it means "will certainly do (that verb).
You may ask what's a difference between しておく shite oku and plain する suru? Both are "will do".
The answer is that any verb ~te+oku means that someone will definitely do that "verb~ておく" right away or in the very near future. It has a "will go ahead and do" kind of feeling to it. And this will be done in a short time.
- 待っておきる matte oku: "I will be waiting (definitely)"
- お弁当を作っておいた obentō o tsukutte oita. (I've made a boxed lunch (for later).)
- 作っておいた tsukutte oita becomes 作っといた tsukuttoita.
- Ron ni denwa shite oku. (I'll call Ron.)
- Mado o akete oku. (I'll open the window.)
- Kasa o katte okimasu. (I'm going to buy an umbrella.)
- Kanojo ni ki o tsukeru you ni itte okimasu. (I'll tell her to be careful.)
- Shukudai o shite okimashita. (I [went ahead and] did my homework.)
- Shukudai shtokimashita.
Te Form + miru, mimasu
In English you say sometimes "I'll see if I can...," meaning that you would a try. Same in Japanese, miru means "to see," and you put the verb in the Te Form adding miru.
みます mimasu same but polite.
- 待ってみる matte miru: "I will see if I can wait"
- Kono kanji o yonde miru. (I'll try to read these kanji.)
- Kono atarashii PC o tsukatte miyou. (Let's give this new PC a try.)
- Sushi o tabete minai no? (Won't you try some sushi?)
- John ni hanashite mimasu. (I'll try to talk to John.)
- Kare ni denwa shite mimashita ga, rusu deshita. (I tried calling him, but he wasn't in.)
~te+oku - "definitely will ~verb".
~te+miru - "will try to~verb".