The number can be expressed as for some integers and . What is the value of ?
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This one took me a while. I kept trying to square the whole thing directly and got lost. The trick is to match it to x² + 3y² - 2xy√3, right?
Yes exactly — that's the standard form for (x+y√3)². Once you see it that way the rest falls into place.
Took me a minute to realize why the cross term is 2xy√3, but yeah this is the way.
Wait, I set x²+3y²=63 and 2xy=36, so xy=18. But that's just the answer without even needing x and y individually? Is that intentional or am I missing something?
Haha same, I solved for x and y separately and then felt silly. But I guess some versions of this don't have such a clean setup.
Medium-hard for me. The 36√3 cross term is what gives it away, but I initially forgot the factor of 2 and got the wrong sign for xy.
The sign is the sneaky part — I had to double check whether y was positive or negative before multiplying.
Is there a reason x and y need to be integers here? Feels like the problem hands you that so you don't worry about other representations.
What it tests
Your handling of square and higher roots, including simplifying radicals.
Common trap
Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.