Determine if the numbers 189 and 289 are perfect squares using prime factorization.

Step 1: Understand the definition of a perfect square using prime factorization.
A number is a perfect square if, in its prime factorization, every prime factor has an even exponent.

Step 2: Perform prime factorization for each given number and determine if it is a perfect square.

• (i) 189:

  $$189 = 3 \times 63 = 3 \times 3 \times 21 = 3 \times 3 \times 3 \times 7 = 3^3 \times 7^1$$

  Since the exponents (3 and 1) are odd, 189 is not a perfect square.

• (ii) 289:

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