What is the value of , where the given expression extends to an infinite number of roots?
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Infinite nested roots always trip me up. I set x = 10√(10√(...)) and got a quadratic, but then I second-guessed the answer. Is that the right approach?
Yeah, that's the standard method. Works because the inner expression is the same as the whole thing.
Don't forget to reject the negative root since we're dealing with a positive value.
I got 100, but I wasn't sure if the expression converges. Does it always converge for any base > 1?
For this specific form, yes. But if the number inside is less than 1, it might diverge or converge to different values.
This one took me longer than expected. I initially tried to compute step by step but that didn't work. Setting up the equation was key.
Nice question! The answer choices are spread out enough that you could also estimate: 10√10 ≈ 31.6, then 10√31.6 ≈ 56, then 10√56 ≈ 75, then 10√75 ≈ 86, approaching 100. But algebra is faster.
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.