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

Question

UpStudy AI · Accepted Answer

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**.
The expression simplifies to approximately 1.0632.

b)

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