Simplify : `( 0.0347 xx 0.0347 xx 0.0347 + (0.9653)^(3))/((0.0347)^(2)-(0.0347)(0.9653)+(0.9653)^(2))`

Options

A. `0.9306`
B. `1.0009`
C. `1.0050`
D. `1`

Answer

To simplify the expression

\[ \frac{(0.0347)^3 + (0.9653)^3}{(0.0347)^2 - (0.0347)(0.9653) + (0.9653)^2} \]

we can follow these steps:

### Step 1: Recognize the Formulas
The numerator \( (0.0347)^3 + (0.9653)^3 \) can be expressed using the sum of cubes formula:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
where \( A = 0.0347 \) and \( B = 0.9653 \).

### Step 2: Apply the Sum of Cubes Formula
Using the formula, we can rewrite the numerator:
\[ (0.0347)^3 + (0.9653)^3 = (0.0347 + 0.9653)((0.0347)^2 - (0.0347)(0.9653) + (0.9653)^2) \]

### Step 3: Simplify the Numerator
Now, calculate \( 0.0347 + 0.9653 \):
\[ 0.0347 + 0.9653 = 1 \]
Thus, the numerator simplifies to:
\[ 1 \cdot ((0.0347)^2 - (0.0347)(0.9653) + (0.9653)^2) \]

### Step 4: Write the Full Expression
Now, substituting back into the original expression, we have:
\[ \frac{1 \cdot ((0.0347)^2 - (0.0347)(0.9653) + (0.9653)^2)}{(0.0347)^2 - (0.0347)(0.9653) + (0.9653)^2} \]

### Step 5: Cancel the Common Terms
The numerator and denominator are now the same, so we can cancel them:
\[ = 1 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{1} \]