If is a positive number and , where the expression extends to an infinite number of roots, what is the value of ?
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Classic infinite radical setup. Let the whole thing equal 10, then notice the inside after the first root is the same expression again. That gives you a simple equation you can solve in like 10 seconds. Answer is 90.
Yep, squaring once and substituting back is the trick. Took me a sec to see it but once you do it's easy.
Wait I got 90 too but I did it by plugging in choices. Is the algebra approach faster?
I always get nervous with these "extends infinitely" problems because it feels like it shouldn't converge, but it does as long as x is positive. Substitution method works perfectly here.
Same, the infinity part used to scare me. Once you realize the tail is identical to the whole thing it's fine.
I initially tried plugging in the choices and wasted time. The substitution is way cleaner: set the expression = 10, then the part under the first radical also = 10. So 10 = sqrt(x + 10), square it, x = 90.
This is the cleanest explanation, thanks. I was overcomplicating it.
Is this really only 555-605? The infinite radical concept trips a lot of people up even if the algebra is simple.
I think the difficulty comes from recognizing the self-similar structure, not the math itself. Once you see it, it's a 500-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.