The value of `root(3)(5+2sqrt(13)) + root(3)(5-2sqrt(13))` is= ……………..

Answer

To solve the expression \( \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}} \), we can follow these steps:

### Step 1: Define the Expression
Let
\[ x = \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}} \]

### Step 2: Cube Both Sides
Now, we will cube both sides to eliminate the cube roots:
\[ x^3 = \left( \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}} \right)^3 \]

### Step 3: Apply the Cube Formula
Using the formula for the cube of a sum, \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), we can expand the right-hand side:
\[ x^3 = \left(5 + 2\sqrt{13}\right) + \left(5 - 2\sqrt{13}\right) + 3 \sqrt[3]{(5 + 2\sqrt{13})(5 - 2\sqrt{13})} \cdot x \]

### Step 4: Simplify the Expression
Calculating \( (5 + 2\sqrt{13})(5 - 2\sqrt{13}) \):
\[ (5 + 2\sqrt{13})(5 - 2\sqrt{13}) = 5^2 - (2\sqrt{13})^2 = 25 - 4 \cdot 13 = 25 - 52 = -27 \]
Thus, we have:
\[ x^3 = 10 + 3 \sqrt[3]{-27} \cdot x \]
Since \( \sqrt[3]{-27} = -3 \), we can substitute:
\[ x^3 = 10 - 9x \]

### Step 5: Rearrange the Equation
Rearranging gives us:
\[ x^3 + 9x - 10 = 0 \]

### Step 6: Factor the Polynomial
To solve the cubic equation, we can try to find rational roots. Testing \( x = 1 \):
\[ 1^3 + 9 \cdot 1 - 10 = 1 + 9 - 10 = 0 \]
Thus, \( x = 1 \) is a root.

### Step 7: Factor Out \( (x - 1) \)
Now, we can factor \( x^3 + 9x - 10 \) as:
\[ (x - 1)(x^2 + x + 10) = 0 \]
The quadratic \( x^2 + x + 10 \) does not have real roots (as its discriminant \( 1 - 40 < 0 \)).

### Step 8: Conclusion
The only real solution is:
\[ x = 1 \]

Thus, the value of \( \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}} \) is:
\[ \boxed{1} \]