?5?7?7?4? = 34

[Tigmafuzz]

use the digits in the new year in various combinations

Judging from your examples, I'm assuming the digits have to stay in order. An easy solution would be 5*7-((7+4)^0) but that's assuming you can add a zero at the end as an exponent. If that's not an option, then the (SPOILER)just as easy solutions would be either:

⌈5√7 + 7 * 4⌉ which is a ceiling function of the 5th root of 7 (≈1.476) plus 7 (which gives us 8.476) multiplied by 4 (which gives us 33.9, and the ceiling function automatically rounds that up to the nearest integer, which is 34.)

or, if you're putting it in a calculator:

⌈5 yroot 7 + 7 * 4⌉ which gives the same result, because the ceiling function is cool like that.
EDIT: if you're putting the previous function in the calculator, you would actually press 7 yroot 5. This function is a different equation entirely that just so happens to also end up with a number between 33 and 34.

Another weird solution, using the floor function instead, would be:

⌊5! / 7 - 7⌋ + 4! which is a floor function of the factorial of 5 (120) divided by 7 (which gives us ≈17.143) minus 7 (leaving us with ≈10.143, which the floor function rounds down to 10) plus the factorial of 4 (which is 24, giving us a total of 34.)

Math is fun.

EDIT2: 5+7+7+4x where x=3.75
lol cheating