1. **State the problem:** Simplify the expression $$b^{\frac{1}{3}} b^{\frac{1}{2}} b^{-\frac{2}{7}}$$ and write the answer using only positive exponents, assuming all variables are positive real numbers. 2. **Recall the exponent rule for products:** When multiplying powers with the same base, add the exponents: $$b^m \cdot b^n = b^{m+n}$$ 3. **Apply the rule:** $$b^{\frac{1}{3}} b^{\frac{1}{2}} b^{-\frac{2}{7}} = b^{\frac{1}{3} + \frac{1}{2} - \frac{2}{7}}$$ 4. **Find a common denominator to add the exponents:** The denominators are 3, 2, and 7. The least common denominator is 42. Convert each fraction: $$\frac{1}{3} = \frac{14}{42}, \quad \frac{1}{2} = \frac{21}{42}, \quad \frac{2}{7} = \frac{12}{42}$$ 5. **Add the exponents:** $$\frac{14}{42} + \frac{21}{42} - \frac{12}{42} = \frac{14 + 21 - 12}{42} = \frac{23}{42}$$ 6. **Write the simplified expression:** $$b^{\frac{23}{42}}$$ 7. **Check the exponent:** It is positive, so no further changes are needed. **Final answer:** $$b^{\frac{23}{42}}$$