`a +b+c= 3, (1)/(a) + (1)/(b) + (1)/(c )=2`
`a^(2) + b^(2) + c^(2)=6` find abc=?

Answer

To solve the problem step by step, we start with the given equations:

1. \( a + b + c = 3 \) (Equation 1)
2. \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 2 \) (Equation 2)
3. \( a^2 + b^2 + c^2 = 6 \) (Equation 3)

We need to find the value of \( abc \).

### Step 1: Rewrite Equation 2
We can rewrite Equation 2 in terms of \( a, b, c \):
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{bc + ac + ab}{abc} = 2 \]
From this, we can express:
\[ bc + ac + ab = 2abc \quad \text{(Equation 4)} \]

### Step 2: Use the identity for squares
We know the identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
Substituting the values from Equations 1 and 3:
\[ 3^2 = 6 + 2(ab + ac + bc) \]
This simplifies to:
\[ 9 = 6 + 2(ab + ac + bc) \]
Subtracting 6 from both sides gives:
\[ 3 = 2(ab + ac + bc) \]
Thus:
\[ ab + ac + bc = \frac{3}{2} \quad \text{(Equation 5)} \]

### Step 3: Substitute Equation 5 into Equation 4
Now we can substitute Equation 5 into Equation 4:
\[ \frac{3}{2} = 2abc \]
Dividing both sides by 2 gives:
\[ abc = \frac{3}{4} \]

### Final Answer
Thus, the value of \( abc \) is:
\[ \boxed{\frac{3}{4}} \]