Solve the Fraction Equation: Simplify (b^7 * b^-4 + b^5) / b^-3

To simplify the problem $\frac{b^7 \cdot b^{-4} + b^5}{b^{-3}}$, we'll begin by applying the rules of exponents.

Step 1: Simplify the multiplication in the numerator.
- Using the product of powers rule, simplify $b^7 \cdot b^{-4}$ as:
$b^7 \cdot b^{-4} = b^{7-4} = b^3$

Step 2: Substitute this back into the expression and rearrange the numerator:
- The expression becomes $\frac{b^3 + b^5}{b^{-3}}$.

Step 3: Simplify the overall expression by applying the quotient of powers rule:
- Distribute the exponent $b^{-3}$ to both terms in the numerator:
$\frac{b^3}{b^{-3}} + \frac{b^5}{b^{-3}}$

Step 4: Using the rule $\frac{a^m}{a^n} = a^{m-n}$, simplify each term:

• $\frac{b^3}{b^{-3}} = b^{3-(-3)} = b^{3+3} = b^6$
• $\frac{b^5}{b^{-3}} = b^{5-(-3)} = b^{5+3} = b^8$

Step 5: Combine these results:
- The final simplified result is $b^6 + b^8$.

Thus, the solution to the expression is $b^6 + b^8$.