If ` (a^(3) + b^(3))/( c^(3)) = (b^(3) + c^(3))/( a^(3)) = (c^(3) + a^(3))/(b^(3))`

Options

A. 1
B. 2
C. 8
D. 27

Answer

To solve the equation

\[ \frac{a^3 + b^3}{c^3} = \frac{b^3 + c^3}{a^3} = \frac{c^3 + a^3}{b^3} \]

we can use the concept of symmetry. This means we can assume that \( a = b = c \) to simplify our calculations.

### Step 1: Assume \( a = b = c \)

Let \( a = b = c = k \) for some constant \( k \).

### Step 2: Substitute into the equation

Now, substituting \( a \), \( b \), and \( c \) with \( k \):

\[ \frac{k^3 + k^3}{k^3} = \frac{k^3 + k^3}{k^3} = \frac{k^3 + k^3}{k^3} \]

### Step 3: Simplify the expressions

Each term simplifies as follows:

\[ \frac{2k^3}{k^3} = \frac{2k^3}{k^3} = \frac{2k^3}{k^3} \]

This simplifies to:

\[ 2 \]

### Step 4: Conclusion

Since all three expressions equal \( 2 \), we conclude that:

\[ \frac{a^3 + b^3}{c^3} = \frac{b^3 + c^3}{a^3} = \frac{c^3 + a^3}{b^3} = 2 \]

Thus, the value is \( 2 \).