Problem 9 · 2026 AMC 8
Medium
Arithmetic & Operations
simplify-radicals
What is the value of this expression?
√16√81√81√16
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Answer: B — 2/3.
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Hint 1 of 2
Top and bottom are built from the very same numbers, 16 and 81, just with their roles swapped. That symmetry says: simplify each piece the same way and the answer should be a clean ratio — both are perfect squares and perfect fourth powers.
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Hint 2 of 2
Work each square root from the inside out: simplify the inner root, multiply, then take the outer root. Or notice 16 = 2⁴ and 81 = 3⁴ and ride the exponents.
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Approach: simplify each nested root from the inside out
- Start with the innermost roots: √81 = 9 and √16 = 4. Now the insides are plain numbers.
- Top inside: 16 × 9 = 144. Bottom inside: 81 × 4 = 324.
- Take the outer roots: √144 = 12 and √324 = 18, so the value is 12/18 = 2/3.
- Why this transfers: nested radicals untangle from the inside out, one layer at a time — never try to swallow the whole tower at once.
Another way — ride the exponents (16 = 2⁴, 81 = 3⁴):
- Write everything in powers of 2 and 3. The top is √(2⁴·3²) = 2²·3, the bottom is √(3⁴·2²) = 3²·2.
- The ratio is (2²·3)/(3²·2) = 2/3. The swapped roles of 16 and 81 are exactly why the answer is just (16/81)1/4 = 2/3.
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