1. **State the problem:** We want to evaluate the expression $$\left[\frac{(1-\alpha) S}{4 \sigma}\right]^{1/4}$$ where $$\alpha = 0.3$$, $$S = 1361\ \text{W m}^{-2}$$, and $$\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2} \text{K}^{-4}$$. 2. **Write the formula:** The expression is $$T = \left[\frac{(1-\alpha) S}{4 \sigma}\right]^{1/4}$$ which is used to calculate the effective temperature of a planet assuming energy balance. 3. **Substitute the values:** $$T = \left[\frac{(1-0.3) \times 1361}{4 \times 5.67 \times 10^{-8}}\right]^{1/4}$$ 4. **Calculate the numerator:** $$(1-0.3) \times 1361 = 0.7 \times 1361 = 952.7$$ 5. **Calculate the denominator:** $$4 \times 5.67 \times 10^{-8} = 2.268 \times 10^{-7}$$ 6. **Form the fraction:** $$\frac{952.7}{2.268 \times 10^{-7}}$$ 7. **Simplify the fraction:** $$\frac{952.7}{2.268 \times 10^{-7}} = 4.201 \times 10^{9}$$ 8. **Take the fourth root:** $$T = (4.201 \times 10^{9})^{1/4}$$ 9. **Calculate the fourth root:** $$T = 255\ \text{Kelvin}$$ **Final answer:** The temperature is $$\boxed{255\ \text{Kelvin}}$$.