Simplify the indices expression: Compute 16^(3/2) + 16^(−3/2) and express your answer as a simplified fraction.

Introduction / Context:
Rational exponents allow us to rewrite powers in terms of roots. 16^(3/2) is the cube of the square root (or the square root of the cube) of 16. Similarly, negative exponents indicate reciprocals. Combine these properties to obtain an exact fractional sum.

Given Data / Assumptions:

• 16^(3/2) and 16^(−3/2).
• 16 = 2^4, √16 = 4.

Concept / Approach:
Compute each term separately: 16^(3/2) = (16^(1/2))^3 = 4^3 = 64. For the negative exponent, 16^(−3/2) = 1 / 16^(3/2) = 1/64. Then add the two values and convert to a single fraction.

Step-by-Step Solution:

Verification / Alternative check:
Compute numerically: 64 + 0.015625 = 64.015625 = 4097/64. Matches.

Why Other Options Are Wrong:

• 0, 1, 16/9097: Incorrect evaluations of the exponents or inversion.
• 65/64: Misses the large 16^(3/2) term (64).

Common Pitfalls:
Confusing 16^(3/2) with (16^3)^(1/2) without carefully evaluating; both are 64, but ensure steps are consistent. Do not forget the reciprocal for the negative exponent.

Final Answer:
4097/64