Simplify : `(0.05xx0.05xx0.05-0.04xx0.04xx0.04)/(0.05xx0.05+0.002+0.04xx0.04)`

Options

A. `1`
B. `0.1`
C. `0.01`
D. `0.001`

Answer

To simplify the expression \((0.05 \times 0.05 \times 0.05 - 0.04 \times 0.04 \times 0.04) / (0.05 \times 0.05 + 0.002 + 0.04 \times 0.04)\), we can follow these steps:

### Step 1: Rewrite the expression using exponents
We can express the terms in the numerator and denominator using exponents:
- \(0.05 \times 0.05 \times 0.05 = (0.05)^3\)
- \(0.04 \times 0.04 \times 0.04 = (0.04)^3\)

So, the numerator becomes:
\[ (0.05)^3 - (0.04)^3 \]

### Step 2: Recognize the difference of cubes
The expression \((a^3 - b^3)\) can be factored using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Let \(a = 0.05\) and \(b = 0.04\). Therefore, we can rewrite the numerator as:
\[ (0.05 - 0.04)((0.05)^2 + (0.05)(0.04) + (0.04)^2) \]

### Step 3: Calculate \(0.05 - 0.04\)
Calculating the difference:
\[ 0.05 - 0.04 = 0.01 \]

### Step 4: Calculate the remaining terms in the numerator
Now we calculate \((0.05)^2 + (0.05)(0.04) + (0.04)^2\):
- \((0.05)^2 = 0.0025\)
- \((0.05)(0.04) = 0.002\)
- \((0.04)^2 = 0.0016\)

Adding these together:
\[ 0.0025 + 0.002 + 0.0016 = 0.0061 \]

### Step 5: Substitute back into the numerator
Now substituting back into the numerator:
\[ 0.01 \times 0.0061 = 0.000061 \]

### Step 6: Calculate the denominator
Now we calculate the denominator:
\[ (0.05 \times 0.05) + 0.002 + (0.04 \times 0.04) \]
Calculating each term:
- \(0.05 \times 0.05 = 0.0025\)
- \(0.04 \times 0.04 = 0.0016\)

Adding these:
\[ 0.0025 + 0.002 + 0.0016 = 0.0061 \]

### Step 7: Final division
Now we divide the numerator by the denominator:
\[ \frac{0.000061}{0.0061} \]

### Step 8: Simplify the fraction
This simplifies to:
\[ \frac{0.000061}{0.0061} = 0.01 \]

Thus, the final simplified result is:
\[ \boxed{0.01} \]