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 of the numbers in the numerator
Calculate each term in the numerator:

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

Now, add these values together:

\[ 0.01 + 0.0001 + 0.000081 = 0.010181 \]

### Step 2: Calculate the squares of the numbers in the denominator
Calculate each term in the denominator:

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

Now, add these values together:

\[ 0.0001 + 0.000001 + 0.00000081 = 0.00010181 \]

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

\[ \frac{0.010181}{0.00010181} \]

### Step 4: Simplify the fraction
To simplify this fraction, we can divide both the numerator and the denominator by \(0.00010181\):

\[ \frac{0.010181 \div 0.00010181}{0.00010181 \div 0.00010181} = \frac{100.0001}{1} \approx 100 \]

### Step 5: Take the square root
Now, we take the square root of the simplified fraction:

\[ \sqrt{100} = 10 \]

### Final Answer
Thus, the value of the expression is

\[ \boxed{10} \]