Evaluate `(97+56sqrt(3))^(1//4)`.

Answer

To evaluate \( (97 + 56\sqrt{3})^{\frac{1}{4}} \), we will follow a systematic approach.

### Step 1: Rewrite the expression
We start with the expression:
\[ 97 + 56\sqrt{3} \]

### Step 2: Identify a suitable form
We want to express \( 97 + 56\sqrt{3} \) in the form of \( (a + b)^2 \). We can set:
\[ a^2 + b^2 + 2ab = 97 + 56\sqrt{3} \]
where \( a^2 + b^2 = 97 \) and \( 2ab = 56\sqrt{3} \).

### Step 3: Solve for \( a \) and \( b \)
From \( 2ab = 56\sqrt{3} \), we can express \( ab \):
\[ ab = 28\sqrt{3} \]

Now we have two equations:
1. \( a^2 + b^2 = 97 \)
2. \( ab = 28\sqrt{3} \)

### Step 4: Use the identity \( (a + b)^2 = a^2 + b^2 + 2ab \)
We can express \( (a + b)^2 \) as:
\[ (a + b)^2 = a^2 + b^2 + 2ab = 97 + 56\sqrt{3} \]

### Step 5: Substitute \( ab \) into the equation
Let’s denote \( s = a + b \) and \( p = ab \). We know:
\[ s^2 = 97 + 56\sqrt{3} \]

### Step 6: Find \( a \) and \( b \)
To find \( a \) and \( b \), we can solve the quadratic equation:
\[ x^2 - sx + p = 0 \]
where \( s = a + b \) and \( p = ab \).

### Step 7: Calculate \( s \)
To find \( s \), we can try different values for \( a \) and \( b \). After some trials, we find:
Let \( a = 7 \) and \( b = 4\sqrt{3} \):
\[ 7^2 + (4\sqrt{3})^2 = 49 + 48 = 97 \]
\[ 2 \cdot 7 \cdot 4\sqrt{3} = 56\sqrt{3} \]

### Step 8: Rewrite the original expression
Thus, we can rewrite:
\[ 97 + 56\sqrt{3} = (7 + 4\sqrt{3})^2 \]

### Step 9: Take the fourth root
Now we can evaluate:
\[ (97 + 56\sqrt{3})^{\frac{1}{4}} = ((7 + 4\sqrt{3})^2)^{\frac{1}{4}} = (7 + 4\sqrt{3})^{\frac{1}{2}} \]

### Step 10: Simplify further
Now we need to simplify \( (7 + 4\sqrt{3})^{\frac{1}{2}} \):
\[ (7 + 4\sqrt{3})^{\frac{1}{2}} = \sqrt{7 + 4\sqrt{3}} \]

### Step 11: Rewrite \( \sqrt{7 + 4\sqrt{3}} \)
We can express \( 7 + 4\sqrt{3} \) as:
\[ \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3} \]

### Final Answer
Thus, the final answer is:
\[ (97 + 56\sqrt{3})^{\frac{1}{4}} = 2 + \sqrt{3} \]