If `a^(2)+b^(2)+c^(2)=250` and ab+bc+ca = `3` , find a+b+c .

Answer

To solve the problem, we need to find the value of \( a + b + c \) given the equations:

1. \( a^2 + b^2 + c^2 = 250 \)
2. \( ab + bc + ca = 3 \)

We can use the identity for the square of a sum:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

### Step 1: Substitute the known values into the identity
Using the values from the problem:

\[ (a + b + c)^2 = 250 + 2 \cdot 3 \]

### Step 2: Calculate \( 2 \cdot 3 \)
Calculate \( 2 \cdot 3 \):

\[ 2 \cdot 3 = 6 \]

### Step 3: Add the values
Now, substitute this back into the equation:

\[ (a + b + c)^2 = 250 + 6 \]

### Step 4: Calculate \( 250 + 6 \)
Now, calculate \( 250 + 6 \):

\[ 250 + 6 = 256 \]

### Step 5: Take the square root
Now we take the square root of both sides to find \( a + b + c \):

\[ a + b + c = \sqrt{256} \]

### Step 6: Calculate \( \sqrt{256} \)
Finally, calculate \( \sqrt{256} \):

\[ \sqrt{256} = 16 \]

### Conclusion
Thus, the value of \( a + b + c \) is:

\[ \boxed{16} \]