If `x^a xx x^b xx x^c = 1` then `a^3 + b^3 + c^3` is equal to :

Options

A. 9
B. abc
C. a + b + c
D. 3abc

Answer

To solve the equation \( x^a \cdot x^b \cdot x^c = 1 \) and find the value of \( a^3 + b^3 + c^3 \), we can follow these steps:

### Step 1: Combine the exponents
Using the property of exponents that states \( x^m \cdot x^n = x^{m+n} \), we can combine the terms on the left side:
\[ x^a \cdot x^b \cdot x^c = x^{a+b+c} \]
So, we rewrite the equation as:
\[ x^{a+b+c} = 1 \]

### Step 2: Analyze the equation
The equation \( x^{a+b+c} = 1 \) holds true for any \( x \) (where \( x \neq 0 \)) if the exponent is zero. Therefore, we set the exponent equal to zero:
\[ a + b + c = 0 \]

### Step 3: Use the identity for cubes
We know from algebra that if \( a + b + c = 0 \), then the sum of cubes can be expressed as:
\[ a^3 + b^3 + c^3 = 3abc \]
This is a standard identity that holds when the sum of the variables is zero.

### Step 4: Conclusion
Thus, we can conclude that:
\[ a^3 + b^3 + c^3 = 3abc \]

### Final Answer
The value of \( a^3 + b^3 + c^3 \) is \( 3abc \).

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