1. **State the problem:** We are given two conditions involving $a$ and $b$ (both greater than 1): $$a^{\frac{1}{a}} = b^{\frac{1}{b}}$$ $$a^{10} = b^{15}$$ We need to find the value of $b$. 2. **Express the first condition in terms of logarithms:** Taking natural logarithms on both sides of the first equation, $$\frac{1}{a} \ln a = \frac{1}{b} \ln b$$ Rearranged, $$\ln a^{\frac{1}{a}} = \ln b^{\frac{1}{b}} \implies \frac{\ln a}{a} = \frac{\ln b}{b}$$ 3. **Express $a$ in terms of $b$ from the second condition:** From $$a^{10} = b^{15}$$, $$a = b^{\frac{15}{10}} = b^{\frac{3}{2}}$$ 4. **Substitute $a = b^{\frac{3}{2}}$ into the logarithmic equation:** $$\frac{\ln (b^{\frac{3}{2}})}{b^{\frac{3}{2}}} = \frac{\ln b}{b}$$ Simplify numerator on left: $$\frac{\frac{3}{2} \ln b}{b^{\frac{3}{2}}} = \frac{\ln b}{b}$$ 5. **Multiply both sides by $b^{\frac{3}{2}} b$ to clear denominators:** $$\frac{3}{2} \ln b \cdot b = \ln b \cdot b^{\frac{3}{2}}$$ 6. **Divide both sides by $\ln b$ (since $b > 1$, $\ln b \neq 0$):** $$\frac{3}{2} b = b^{\frac{3}{2}}$$ 7. **Rewrite $b^{\frac{3}{2}}$ as $b \cdot b^{\frac{1}{2}}$ and divide both sides by $b$:** $$\frac{3}{2} = b^{\frac{1}{2}}$$ 8. **Now solve for $b$:** $$b = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ **Final Answer:** $$b = \frac{9}{4}$$