`((0.05)^(2)+(0.41)^(2)+(0.073)^(2))/((0.005)^(2)+(0.041)^(2)+(0.0073)^(2))` is

Options

A. `10`
B. `100`
C. `1000`
D. None of these

Answer

To solve the expression \(\frac{(0.05)^{2}+(0.41)^{2}+(0.073)^{2}}{(0.005)^{2}+(0.041)^{2}+(0.0073)^{2}}\), we can follow these steps:

### Step 1: Rewrite the numbers in terms of powers of 10
We can express the decimals as:
- \(0.05 = 5 \times 10^{-2}\)
- \(0.41 = 4.1 \times 10^{-1}\)
- \(0.073 = 7.3 \times 10^{-2}\)
- \(0.005 = 5 \times 10^{-3}\)
- \(0.041 = 4.1 \times 10^{-2}\)
- \(0.0073 = 7.3 \times 10^{-3}\)

### Step 2: Substitute these values into the expression
The expression now becomes:
\[ \frac{(5 \times 10^{-2})^{2} + (4.1 \times 10^{-1})^{2} + (7.3 \times 10^{-2})^{2}}{(5 \times 10^{-3})^{2} + (4.1 \times 10^{-2})^{2} + (7.3 \times 10^{-3})^{2}} \]

### Step 3: Simplify the squares
Calculating the squares:
- Numerator:
- \((5 \times 10^{-2})^{2} = 25 \times 10^{-4}\)
- \((4.1 \times 10^{-1})^{2} = 16.81 \times 10^{-2}\)
- \((7.3 \times 10^{-2})^{2} = 53.29 \times 10^{-4}\)

- Denominator:
- \((5 \times 10^{-3})^{2} = 25 \times 10^{-6}\)
- \((4.1 \times 10^{-2})^{2} = 16.81 \times 10^{-4}\)
- \((7.3 \times 10^{-3})^{2} = 53.29 \times 10^{-6}\)

### Step 4: Combine the terms in the numerator and denominator
Now, we can combine the terms:
- Numerator:
\[ 25 \times 10^{-4} + 16.81 \times 10^{-2} + 53.29 \times 10^{-4} = 25 \times 10^{-4} + 168.1 \times 10^{-4} + 53.29 \times 10^{-4} = 246.39 \times 10^{-4} \]

- Denominator:
\[ 25 \times 10^{-6} + 16.81 \times 10^{-4} + 53.29 \times 10^{-6} = 25 \times 10^{-6} + 168.1 \times 10^{-6} + 53.29 \times 10^{-6} = 246.39 \times 10^{-6} \]

### Step 5: Write the expression in simplified form
Now we have:
\[ \frac{246.39 \times 10^{-4}}{246.39 \times 10^{-6}} = \frac{246.39}{246.39} \times 10^{2} = 10^{2} = 100 \]

### Conclusion
Thus, the value of the expression is \(100\).