The expression shown is $8^{-7} \cdot 8^3$, and the goal is to rewrite this using a single positive exponent.

We can apply the product of powers property of exponents, which states that when multiplying two expressions with the same base, we add their exponents: $a^m \cdot a^n = a^{m+n}$

Applying this to $8^{-7} \cdot 8^3$, we get: $8^{-7 + 3} = 8^{-4}$

To express this with a positive exponent, we use the property that $a^{-n} = \frac{1}{a^n}$: $8^{-4} = \frac{1}{8^4}$

Thus, the expression rewritten with a single positive exponent is: $\frac{1}{8^4}$

Would you like more details on this process?

Here are five related questions to deepen your understanding:

1. How do you simplify expressions with negative exponents?
2. What are the rules for dividing exponents with the same base?
3. How do you simplify expressions involving fractional exponents?
4. What happens when you raise a power to another power in exponents?
5. How can you apply the product of powers property to variables, not just numbers?

Tip: Remember, negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent.