Let's solve the equation step by step:

The given expression is:

$3 \cdot 4^{-2} + 9 \cdot 8^{-1}$

Step 1: Simplify $4^{-2}$

Using the rule of negative exponents $(a^{-n} = \frac{1}{a^n})$: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

So the expression becomes: $3 \cdot \frac{1}{16} + 9 \cdot 8^{-1}$

Step 2: Simplify $8^{-1}$

Using the rule of negative exponents again: $8^{-1} = \frac{1}{8}$

So the expression now is: $3 \cdot \frac{1}{16} + 9 \cdot \frac{1}{8}$

Step 3: Simplify the multiplications

$3 \cdot \frac{1}{16} = \frac{3}{16}, \quad 9 \cdot \frac{1}{8} = \frac{9}{8}$

The expression becomes: $\frac{3}{16} + \frac{9}{8}$

Step 4: Find a common denominator

The least common denominator (LCD) of 16 and 8 is 16. Rewrite $\frac{9}{8}$ with denominator 16: $\frac{9}{8} = \frac{18}{16}$

Now the expression is: $\frac{3}{16} + \frac{18}{16}$

Step 5: Add the fractions

$\frac{3}{16} + \frac{18}{16} = \frac{3 + 18}{16} = \frac{21}{16}$

Final Answer:

$\boxed{\frac{21}{16}}$

Let me know if you'd like additional details or explanations!

Here are 5 related questions for deeper understanding:

1. What is the general rule for simplifying negative exponents?
2. How can you identify the least common denominator for two fractions?
3. How do you add fractions with different denominators?
4. What is the result of $4^{-3}$ in fraction form?
5. How does changing the order of operations affect calculations?

Tip: When working with negative exponents, always rewrite them as positive exponents first for clarity!