What is the value of ( 197^2 ) using an identity?

Correct answer: (38809)

The direct answer is option A: 38809. Choose a nearby round number: 197 = 200 - 3. Use (a - b)^2 = a^2 - 2ab + b^2. Thus, 197^2 = (200 - 3)^2 = 200^2 - 2(200)(3) + 3^2 = 40000 - 1200 + 9 = 38809. Option A is correct because the calculation gives 38809. Option B, 38800, is close but leaves out the final +9 or uses an incomplete middle-term calculation. Option C, 39409, is too large and does not follow the subtraction-square identity; using 200 + 3 would give 203^2, not 197^2. Option D, 39809, is also incorrect and does not result from the correct three-term expansion. Another check is ordinary multiplication: 197 × 197 = 197 × (200 - 3) = 39400 - 591 = 38809. The important point is that a number just below a convenient round number can be squared quickly by subtracting twice the product and adding the small square. Memory cue: for (near number - small number)^2, square the near number, subtract the double product, then add the small square.