Let's break down the given expression step by step and simplify it.
The given expression is:
[tex]\[
\frac{1}{3} \sqrt{\frac{\pi}{r^4}} \times 6 \sqrt{\frac{1}{7}}
\][/tex]
### Step 1: Simplify the fraction multiplications
First, consider the constants outside of the square roots:
[tex]\[
\frac{1}{3} \times 6
\][/tex]
This simplifies to:
[tex]\[
2
\][/tex]
### Step 2: Simplify inside the square roots
Now, look at the square root terms separately.
For the first square root:
[tex]\[
\sqrt{\frac{\pi}{r^4}}
\][/tex]
This can be rewritten as:
[tex]\[
\sqrt{\pi} \times \sqrt{r^{-4}}
\][/tex]
Which further simplifies to:
[tex]\[
\sqrt{\pi} \times \frac{1}{r^2}
\][/tex]
For the second square root:
[tex]\[
\sqrt{\frac{1}{7}}
\][/tex]
This is equivalent to:
[tex]\[
\frac{1}{\sqrt{7}}
\][/tex]
### Step 3: Combine the simplified parts
Now, multiply these simplified parts together:
[tex]\[
2 \times \sqrt{\pi} \times \frac{1}{r^2} \times \frac{1}{\sqrt{7}}
\][/tex]
### Step 4: Combine the fractions
Combine the fractions:
[tex]\[
2 \times \sqrt{\pi} \times \frac{1}{r^2 \sqrt{7}}
\][/tex]
This can be rewritten as:
[tex]\[
2 \times \sqrt{\pi} \times \frac{1}{r^2 \times \sqrt{7}}
\][/tex]
### Step 5: Final combination
Combine all the parts into a single fraction:
[tex]\[
\frac{2 \sqrt{\pi}}{r^2 \sqrt{7}}
\][/tex]
### Step 6: Simplify the numerical constants
We can simplify the numerical part:
[tex]\[
\frac{2}{\sqrt{7}} \approx 0.755928946018454
\][/tex]
Therefore, combining this constant with [tex]\( \sqrt{\pi} \)[/tex] and [tex]\( r^{-2} \)[/tex], we get:
[tex]\[
0.755928946018454 \times \sqrt{\pi} \times r^{-2} \times \frac{1}{r^2}
\][/tex]
Simplifying further, we get:
[tex]\[
0.755928946018454 \sqrt{\pi} r^{-4}
\][/tex]
or equivalently:
[tex]\[
0.755928946018454 \sqrt{\pi} \sqrt{r^{-4}}
\][/tex]
Thus, the final simplified result is:
[tex]\[
0.755928946018454 \sqrt{\pi} \sqrt{r^{-4}}
\][/tex]
This is the simplified form of the given expression.
