1. The problem is to evaluate the expression $86^{1\frac{1}{2}}$, which means $86$ raised to the power of $1.5$ or $\frac{3}{2}$. 2. Recall the rule for fractional exponents: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$. 3. Rewrite the exponent: $1\frac{1}{2} = \frac{3}{2}$. 4. So, $86^{1\frac{1}{2}} = 86^{\frac{3}{2}} = (86^{\frac{1}{2}})^3 = (\sqrt{86})^3$. 5. Calculate $\sqrt{86} \approx 9.2736$. 6. Then cube it: $(9.2736)^3 = 9.2736 \times 9.2736 \times 9.2736 \approx 797.664$. 7. Therefore, $86^{1\frac{1}{2}} \approx 797.664$. This completes the evaluation of the first problem.