If `(1)/(a+b+c)=(1)/(a)+(1)/(b)+(1)/(c)` then find `(1)/(a^(7))+(1)/(b^(6))+(1)/(c^(8))`

Options

A. 1
B. 0
C. 3
D. 2

Answer

To solve the equation \(\frac{1}{a+b+c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\) and find \(\frac{1}{a^7} + \frac{1}{b^6} + \frac{1}{c^8}\), we can follow these steps:

### Step 1: Understand the given equation
We start with the equation:

\[ \frac{1}{a+b+c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \]

This implies that the sum of the reciprocals of \(a\), \(b\), and \(c\) equals the reciprocal of their sum.

### Step 2: Find suitable values for \(a\), \(b\), and \(c\)
To satisfy the equation, we can choose values for \(a\), \(b\), and \(c\). Let's try:

- \(a = 1\)
- \(b = 1\)
- \(c = -1\)

### Step 3: Verify the chosen values
Now we check if these values satisfy the equation:

1. Calculate \(a + b + c\):
\[ a + b + c = 1 + 1 - 1 = 1 \]

2. Calculate \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\):
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1} + \frac{1}{1} + \frac{1}{-1} = 1 + 1 - 1 = 1 \]

Since both sides are equal, our values are valid.

### Step 4: Calculate \(\frac{1}{a^7} + \frac{1}{b^6} + \frac{1}{c^8}\)
Now we substitute these values into the expression we need to find:

\[ \frac{1}{a^7} + \frac{1}{b^6} + \frac{1}{c^8} \]

Substituting \(a = 1\), \(b = 1\), and \(c = -1\):

\[ \frac{1}{1^7} + \frac{1}{1^6} + \frac{1}{(-1)^8} \]

Calculating each term:

1. \(\frac{1}{1^7} = 1\)
2. \(\frac{1}{1^6} = 1\)
3. \(\frac{1}{(-1)^8} = \frac{1}{1} = 1\)

### Step 5: Sum the results
Now we sum these values:

\[ 1 + 1 + 1 = 3 \]

Thus, the final answer is:

\[ \frac{1}{a^7} + \frac{1}{b^6} + \frac{1}{c^8} = 3 \]

### Conclusion
The answer is \(3\).