Sure, let's solve the given expression step by step.
We are given the expression:
[tex]\[ 0.1 \times 8 \sqrt{\frac{1}{10^9}} \][/tex]
### Step 1: Simplify the term inside the square root
First, let's start with the term inside the square root:
[tex]\[ \frac{1}{10^9} \][/tex]
This simplifies to:
[tex]\[ 1 \times 10^{-9} \][/tex]
### Step 2: Calculate the square root
Next, we need to find the square root of [tex]\( 1 \times 10^{-9} \)[/tex]:
[tex]\[ \sqrt{1 \times 10^{-9}} \][/tex]
The square root of [tex]\( 10^{-9} \)[/tex] is [tex]\( 10^{-4.5} \)[/tex], since:
[tex]\[ \sqrt{10^{-9}} = 10^{-4.5} \][/tex]
Multiplying:
[tex]\[ \sqrt{1 \times 10^{-9}} = 1 \times 10^{-4.5} \][/tex]
Numerically this equates to:
[tex]\[ \sqrt{1 \times 10^{-9}} \approx 3.1622776601683795 \times 10^{-5} \][/tex]
### Step 3: Perform the multiplications
Now, we multiply this result by 8:
[tex]\[ 8 \times 3.1622776601683795 \times 10^{-5} \][/tex]
[tex]\[ \approx 2.5298221281347037 \times 10^{-4} \][/tex]
Finally, multiply by 0.1:
[tex]\[ 0.1 \times 2.5298221281347037 \times 10^{-4} \][/tex]
[tex]\[ \approx 2.5298221281347037 \times 10^{-5} \][/tex]
### Summary
Putting it all together, we find:
[tex]\[ 0.1 \times 8 \sqrt{\frac{1}{10^9}} \approx 2.5298221281347037 \times 10^{-5} \][/tex]
So, the detailed solution results in:
[tex]\[ 2.5298221281347037 \times 10^{-5} \][/tex]
