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TMUA 2023 · Paper 2 · Question 3 of 20

TMUA 2023 Paper 2 Question 3

Proof and counterexample — Counterexamples to an index law. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Proof and counterexampleCounterexamples to an index law8 options
Consider the claim:

For all positive real numbers x and y, √xy = x√y

Which of the following is/are a counterexample to the claim?

I   x = 1, y = 16
II  x = 2, y = 8
III x = 3, y = 4

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · C
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
In I, both sides are 1, since every power of 1 is 1. In III, √34 = √81 = 9 and 3√4 = 32 = 9, so they agree. In II, √28 = √256 = 16 while 2√8 = 22√2 ≈ 7.1, so they differ. Only II is a counterexample.