The product of 1.7 x 104 and 12.5 x 10-6 is expressed in the standard form as…

Concept: To multiply numbers written as a x 10m, multiply the coefficients and add the powers of 10: (a x 10m) x (b x 10n) = (a x b) x 10(m+n). A number is in standard (scientific) form k x 10n only when its coefficient satisfies 1 <= k < 10.

1. Multiply the coefficients: 1.7 x 12.5 = 21.25

2. Add the powers of 10: 104 x 10-6 = 10(4 + (-6)) = 10-2

3. Combine the two results: the product is 21.25 x 10-2

4. 21.25 is not between 1 and 10, so rewrite it in standard form: 21.25 = 2.125 x 101, which makes the product 2.125 x 101 x 10-2 = 2.125 x 10-1

5. Reading off the standard form k x 10n gives k = 2.125 and n = -1

6. Compute 2k + n = 2 x 2.125 + (-1) = 4.25 - 1 = 3.25

Cross-check: Evaluating directly, 1.7 x 104 = 17000 and 12.5 x 10-6 = 0.0000125; multiplying gives 17000 x 0.0000125 = 0.2125 = 2.125 x 10-1, the same k and n found above, confirming 2k + n = 3.25.