Sure! Let's simplify the given expression step-by-step.
We start with the expression:
[tex]\[
\frac{\frac{3 r s^2}{14 t^2 u}}{\frac{6 s^5}{7 t^5 u^3}}
\][/tex]
When we divide by a fraction, it's equivalent to multiplying by its reciprocal. So, we rewrite the expression as:
[tex]\[
\frac{3 r s^2}{14 t^2 u} \times \frac{7 t^5 u^3}{6 s^5}
\][/tex]
Now, let's multiply the numerators together and the denominators together:
Numerator:
[tex]\[
3 r s^2 \times 7 t^5 u^3 = 21 r s^2 t^5 u^3
\][/tex]
Denominator:
[tex]\[
14 t^2 u \times 6 s^5 = 84 t^2 u s^5
\][/tex]
This gives us:
[tex]\[
\frac{21 r s^2 t^5 u^3}{84 t^2 u s^5}
\][/tex]
Next, we simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor.
First, simplify the coefficients:
[tex]\[
\frac{21}{84} = \frac{1}{4}
\][/tex]
Now, let's simplify the variables one by one:
For [tex]\( t \)[/tex]:
[tex]\[
\frac{t^5}{t^2} = t^{5-2} = t^3
\][/tex]
For [tex]\( u \)[/tex]:
[tex]\[
\frac{u^3}{u} = u^{3-1} = u^2
\][/tex]
For [tex]\( s \)[/tex]:
[tex]\[
\frac{s^2}{s^5} = s^{2-5} = s^{-3} = \frac{1}{s^3}
\][/tex]
Putting all these together:
[tex]\[
\frac{21 r s^2 t^5 u^3}{84 t^2 u s^5} = \frac{1}{4} \cdot r \cdot t^3 \cdot u^2 \cdot \frac{1}{s^3} = \frac{r t^3 u^2}{4 s^3}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{r t^3 u^2}{4 s^3}
\][/tex]
