If `1^(3) + 2^(3)`+……. +`9^(3)`=2025, then the approximate value of `(0.11)^(3)+(0.22)^(3)`+…..+`(0.99)^(3)` is

Options

A. 0.2695
B. 0.3695
C. 2.695
D. 3.695

Answer

To solve the problem, we need to find the approximate value of the sum \( (0.11)^3 + (0.22)^3 + \ldots + (0.99)^3 \).

### Step 1: Recognize the pattern
The terms in the sum can be rewritten as:
\[ (0.11)^3 + (0.22)^3 + (0.33)^3 + (0.44)^3 + (0.55)^3 + (0.66)^3 + (0.77)^3 + (0.88)^3 + (0.99)^3 \]
This can be expressed as:
\[ (0.11)^3 \cdot (1^3 + 2^3 + \ldots + 9^3) \]

### Step 2: Factor out \( (0.11)^3 \)
We know from the problem statement that:
\[ 1^3 + 2^3 + \ldots + 9^3 = 2025 \]
Thus, we can factor out \( (0.11)^3 \):
\[ (0.11)^3 \cdot 2025 \]

### Step 3: Calculate \( (0.11)^3 \)
Now, we calculate \( (0.11)^3 \):
\[ (0.11)^3 = 0.11 \times 0.11 \times 0.11 = 0.001331 \]

### Step 4: Multiply by 2025
Now we multiply \( 0.001331 \) by \( 2025 \):
\[ 0.001331 \times 2025 \]

### Step 5: Perform the multiplication
Calculating this gives:
\[ 0.001331 \times 2025 \approx 2.695 \]

### Conclusion
Thus, the approximate value of \( (0.11)^3 + (0.22)^3 + \ldots + (0.99)^3 \) is approximately \( 2.695 \).

### Final Answer
The final answer is approximately \( 2.695 \).

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