2026 AMC 8 Problem 9: Solution

Work through 2026 AMC 8 Problem 9 on radicals with a written solution, hints, and a visual explanation.

AlgebraLevel 2 · Core skill

What is the value of the expression shown below?

The square root of (16 times the square root of 81), divided by the square root of (81 times the square root of 16).
Diagram 1
  • A. 4/9
  • B. 2/3
  • C. 1
  • D. 3/2
  • E. 9/4
Hints
  1. Simplify the innermost square roots first, working from the inside out.
  2. Compute √81 and √16 individually before multiplying under the outer radicals.
  3. Once you have clean numbers under each outer square root, simplify the numerator and denominator separately, then divide.
Read the step-by-step solution

Answer: B · 2/3

  1. Simplify the numerator

    √81 = 9, so 16 × 9 = 144. √144 = 12.

    √(16 × 9) = √144 = 12

  2. Simplify the denominator

    √16 = 4, so 81 × 4 = 324. √324 = 18.

    √(81 × 4) = √324 = 18

  3. Divide

    12/18 = 2/3.

    12/18 = 2/3

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