To simplify the expressions provided in the image:

Problem A: $7^{-8} \cdot 7^6$

• Using the rule of exponents, $a^m \cdot a^n = a^{m+n}$: $7^{-8} \cdot 7^6 = 7^{-8+6} = 7^{-2}.$
• Simplify $7^{-2}$ into a positive exponent: $7^{-2} = \frac{1}{7^2} = \frac{1}{49}.$

Answer for A: $\frac{1}{49}$

Problem B: $3 \cdot 3^7$

• Again using the rule of exponents, rewrite $3$ as $3^1$: $3 \cdot 3^7 = 3^1 \cdot 3^7 = 3^{1+7} = 3^8.$
• The simplified result is: $3^8.$

Answer for B: $3^8$ or $6561$ (if expanded).

Let me know if you'd like a step-by-step explanation or further assistance! 😊

Follow-Up Questions:

1. What are the basic rules of exponents, such as addition and subtraction?
2. How do negative exponents convert to fractions?
3. Can you explain why $a^0 = 1$ for any nonzero $a$?
4. How would the rules differ if the bases in these expressions were different?
5. What is the process for evaluating $3^8$ without a calculator?

Tip: Always simplify exponents step by step to avoid mistakes, and rewrite negative exponents to ensure clarity!