Meaning

“Damemoto de _______ suru” is used when you think something probably can’t be done, but there is nothing to lose if you go for it. It expresses the act of doing something with little expectation. “Damemoto” is a contraction of “dame de moto moto.” “Dame de moto moto” is sometimes used instead of “damemoto.”

Japanese Example
A：今年の夏に、できれば一週間休みを取って海外旅行を
したいんだけど、部長が何て言うか心配…。
B：いいって言うかもしれないでしょ。
ダメもとで聞いてみなよ。

A：Kotoshi no natsu ni、dekireba isshukan yasumi o totte kaigairyokou wo shitaindakedo buchou ga nan’te iu ka shinpai…
B：Iitte iu kamo shirenai desho.  Damemoto de kiite minayo!

English Translation
A: I want to see if I can take a week off this summer to travel abroad but I’m not really sure what to say to my boss…
B: He might say yes. Just ask–you’ve got nothing to lose.

Would you like to try to translate the sentence below?
 – I tried asking this girl I like out on a date because I had nothing to lose.