1. **State the problem:** Find the exact value of $\frac{3}{8} \div \left( \frac{5}{12} + \frac{1}{3} \right)$. 2. **Simplify inside the parentheses:** Convert $\frac{1}{3}$ to have a common denominator with $\frac{5}{12}$. Since $12$ is a multiple of $3$, write $\frac{1}{3} = \frac{4}{12}$. 3. **Add the fractions in the denominator:** $$\frac{5}{12} + \frac{4}{12} = \frac{9}{12}.$$ 4. **Simplify the fraction:** $\frac{9}{12} = \frac{3}{4}$ after dividing numerator and denominator by $3$. 5. **Rewrite the division problem:** $$\frac{3}{8} \div \frac{3}{4}.$$ 6. **Recall that dividing by a fraction is multiplying by its reciprocal:** $$\frac{3}{8} \times \frac{4}{3}.$$ 7. **Multiply numerators and denominators:** $$\frac{3 \times 4}{8 \times 3} = \frac{12}{24}.$$ 8. **Simplify the fraction:** $$\frac{12}{24} = \frac{1}{2}.$$ **Final answer:** The exact value is $\boxed{\frac{1}{2}}$.