Problem 3 · 2017 AMC 8
Medium
Arithmetic & Operations
perfect-square
What is the value of the expression √(16 · √(8 · √4)) ?
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Answer: C — 8.
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Hint 1 of 2
Nested radicals only untangle from the inside out — you can't simplify the outer √ until you know what's under it. Start at the deepest √4.
Still stuck? Show hint 2 →
Hint 2 of 2
Watch for a chain reaction: each square root turns into a whole number, which then makes the next product a perfect square too. That's the design of these problems.
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Approach: peel from the inside out
- Start at the deepest layer: √4 = 2. This is the whole game — it makes the next product land on a perfect square.
- Middle layer: √(8 · 2) = √16 = 4. Again a perfect square pops out.
- Outer layer: √(16 · 4) = √64 = 8.
- Why it works: contest nested radicals are built so every layer collapses to an integer. If yours doesn't, recheck the layer below.
Another way — exponents instead of roots:
- Write each √ as a power of 1/2: the expression is (16 · (8 · 4^(1/2))^(1/2))^(1/2).
- Innermost 4^(1/2) = 2, then 8·2 = 16 and 16^(1/2) = 4, then 16·4 = 64 and 64^(1/2) = 8 — same collapse, viewed as halving exponents.
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