If and , then what is the value of ?
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Finally got this one after staring at it for 10 minutes. The trick is to realize that 21+12√3 can be written as (something + something√3)². I won't spoil it, but once you find that, the rest falls out. Hard but satisfying!
Yeah, I was stuck until I tried to express it as (x + y√3)² and solve for x and y. Then it clicked.
I'm confused: after simplifying the inner radical, I get 3+2√3, but then I have √(1 + 3+2√3) = √(4+2√3). Can someone explain how to simplify that? I know the final answer should be a perfect square, but I'm not seeing it.
Think about writing 4+2√3 as (√a + √b)² again. It's the same pattern. Expand and match terms.
Oh! So it's like a nested radical simplification. I'll try that.
This is a tricky one. The condition a < b makes it a bit easier to pick the right values. I got a=1, but I won't say if that's right. Definitely a 700-level question.
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.