The value of `sqrt(((0.1)^(2) + (0.01)^(2) + (0.009)^(2))/((0.01)^(2) + (0.001)^(2) + (0.0009)^(2)))` is :

Options

A. `10^(2)`
B. 10
C. 0.1
D. 0.01

Answer

To solve the expression

\[ \sqrt{\frac{(0.1)^2 + (0.01)^2 + (0.009)^2}{(0.01)^2 + (0.001)^2 + (0.0009)^2}}, \]

we will follow these steps:

### Step 1: Calculate the squares in the numerator
First, we calculate each term in the numerator:

- \((0.1)^2 = 0.01\)
- \((0.01)^2 = 0.0001\)
- \((0.009)^2 = 0.000081\)

Now, we sum these values:

\[ 0.01 + 0.0001 + 0.000081 = 0.010181. \]

### Step 2: Calculate the squares in the denominator
Next, we calculate each term in the denominator:

- \((0.01)^2 = 0.0001\)
- \((0.001)^2 = 0.000001\)
- \((0.0009)^2 = 0.00000081\)

Now, we sum these values:

\[ 0.0001 + 0.000001 + 0.00000081 = 0.00010081. \]

### Step 3: Form the fraction
Now we can form the fraction:

\[ \frac{0.010181}{0.00010081}. \]

### Step 4: Simplify the fraction
To simplify the fraction, we can multiply both the numerator and the denominator by \(100000\) to eliminate the decimal points:

\[ \frac{0.010181 \times 100000}{0.00010081 \times 100000} = \frac{1018.1}{10.081}. \]

### Step 5: Perform the division
Now we perform the division:

\[ \frac{1018.1}{10.081} \approx 101.5. \]

### Step 6: Take the square root
Finally, we take the square root of the result:

\[ \sqrt{101.5} \approx 10. \]

Thus, the value of the original expression is approximately \(10\).

### Final Answer:
The value of the expression is \(10\).

---