Evaluate: 16$^{0.16}$ × 16$^{0.04}$ × 2$^{0.2}$

Analyze the question: All the terms can be written with the same base, so we can add the indices.

Concept: When multiplying powers with the same base, add the indices: a^m × a^n = a^(m+n). Also, 16 = 2⁴.

Solution:

1. Combine the two powers of 16: 16^0.16 × 16^0.04 = 16^(0.16 + 0.04) = 16^0.2

2. Write 16 as a power of 2: 16^0.2 = (2⁴)^0.2 = 2^(4 × 0.2) = 2^0.8

3. Multiply by the last term: 2^0.8 × 2^0.2 = 2^(0.8 + 0.2) = 2^1 = 2

Answer: 2, which is option A.

Check against the options: Option C, 2⁰, equals 1, so it is not correct. Option B (0) and option D (1/2) do not match either.

Tip: In index questions, first convert everything to the same base (here, base 2), then add or multiply the indices. Notice that 0.16 + 0.04 = 0.2 and 4 × 0.2 = 0.8 gave an easy 0.8 + 0.2 = 1 at the end. Nice numbers like this are a sign you are on the right track.