We need to evaluate:

$16^{-\frac{5}{4}}$

Step 1: Express 16 as a Power of 2

Since $16 = 2^4$, we rewrite the expression:

$(2^4)^{-\frac{5}{4}}$

Step 2: Apply Exponent Rule $(a^m)^n = a^{m \cdot n}$

Multiplying the exponents:

$2^{4 \times \left(-\frac{5}{4}\right)}$

$2^{-5}$

Step 3: Evaluate the Negative Exponent

Using the property $a^{-n} = \frac{1}{a^n}$:

$2^{-5} = \frac{1}{2^5}$

$\frac{1}{32}$

Final Answer:

$\frac{1}{32}$

Would you like further clarification or a different approach?

Related Questions:

1. How do you simplify expressions with fractional exponents?
2. What are the exponent rules, and how do they work?
3. How do you evaluate expressions with negative exponents?
4. How do you convert between radical and exponent notation?
5. What are some real-world applications of exponent rules?

Tip:

A negative exponent means "reciprocal," and a fractional exponent indicates a root operation!