The problem to solve is:

$16 \cdot \left(\frac{2^{-3}}{2^3}\right)$

Step-by-Step Solution:

1. Simplify the fraction inside the parentheses: Using the rule of exponents, $\frac{a^m}{a^n} = a^{m-n}$: $\frac{2^{-3}}{2^3} = 2^{-3-3} = 2^{-6}.$

2. Substitute back into the expression: $16 \cdot \left(2^{-6}\right).$

3. Simplify further: Recall that $a^{-n} = \frac{1}{a^n}$: $2^{-6} = \frac{1}{2^6}.$ So: $16 \cdot \frac{1}{2^6}.$

4. Evaluate $2^6$: $2^6 = 64.$ This simplifies the expression to: $16 \cdot \frac{1}{64} = \frac{16}{64}.$

5. Simplify the fraction: $\frac{16}{64} = \frac{1}{4}.$

Final Answer:

$\boxed{\frac{1}{4}}$

Would you like further clarification or a deeper explanation of any step? 😊

Related Questions:

1. What are the laws of exponents and how do they work?
2. How do negative exponents affect a base?
3. How can you simplify expressions with powers of fractions?
4. What are practical applications of simplifying exponential expressions?
5. How does fractional arithmetic work in general?

Tip:

Always remember that a negative exponent simply means "reciprocal of the positive power."