1. **State the problem:** Given the expression $$\sqrt[5]{b^7} = b^m$$, find the value of the exponent $$m$$. 2. **Raise both sides to the power of 5:** $$\left(\sqrt[5]{b^7}\right)^5 = (b^m)^5$$ 3. **Use the definition of the fifth root:** $$\left(\sqrt[5]{b^7}\right)^5 = b^7$$ 4. **Apply the power of a power property:** $$(b^m)^5 = b^{5m}$$ 5. **Equate the expressions:** $$b^7 = b^{5m}$$ 6. **Since the bases are the same, equate the exponents:** $$7 = 5m$$ 7. **Solve for $$m$$:** $$m = \frac{7}{5}$$ 8. **Substitute back:** $$\sqrt[5]{b^7} = b^{\frac{7}{5}}$$