b) \( \sqrt[\sqrt{5}+\sqrt{2}+\left(\sqrt{2}\right)^{2}]{\sqrt{2}} \)

To evaluate the expression: \[ \sqrt[\sqrt{5} + \sqrt{2} + \left(\sqrt{2}\right)^2]{\sqrt{2}} \] let's break it down step by step. ### Step 1: Simplify the Denominator First, simplify the exponent denominator: \[ \sqrt{5} + \sqrt{2} + \left(\sqrt{2}\right)^2 \] Since \(\left(\sqrt{2}\right)^2 = 2\), the denominator becomes: \[ \sqrt{5} + \sqrt{2} + 2 \] ### Step 2: Rewrite the Expression Using Exponents The original expression can be rewritten using exponent notation: \[ \sqrt{2}^{\frac{1}{\sqrt{5} + \sqrt{2} + 2}} \] Recall that \(\sqrt{2} = 2^{1/2}\), so: \[ 2^{\frac{1}{2(\sqrt{5} + \sqrt{2} + 2)}} \] ### Step 3: Approximate the Numerical Value To find a numerical approximation: 1. **Calculate the Denominator:** \[ \sqrt{5} \approx 2.23607,\quad \sqrt{2} \approx 1.41421 \] \[ \sqrt{5} + \sqrt{2} + 2 \approx 2.23607 + 1.41421 + 2 = 5.65028 \] 2. **Compute the Exponent:** \[ \frac{1}{2 \times 5.65028} = \frac{1}{11.30056} \approx 0.0885 \] 3. **Evaluate \(2^{0.0885}\):** \[ 2^{0.0885} \approx 1.0632 \] ### Final Answer The expression simplifies to approximately **1.0632**.