Certainly! Let's simplify the given complex fraction step-by-step.
### Step 1: Represent the Fraction
We start by writing the given complex fraction as follows:
[tex]\[
\frac{\frac{3 t^5 u^3}{28 s^3}}{\frac{6 t^2 u}{7 r s^2}}
\][/tex]
### Step 2: Invert and Multiply the Denominator
To simplify a fraction of fractions, we can multiply by the reciprocal of the denominator:
[tex]\[
\frac{3 t^5 u^3}{28 s^3} \times \frac{7 r s^2}{6 t^2 u}
\][/tex]
### Step 3: Simplify the Multiplication
Next, we multiply the numerators together and the denominators together:
[tex]\[
\frac{(3 t^5 u^3) \cdot (7 r s^2)}{(28 s^3) \cdot (6 t^2 u)}
\][/tex]
### Step 4: Combine and Rearrange the Terms
Combining all terms, we get:
[tex]\[
\frac{3 t^5 u^3 \cdot 7 r s^2}{28 s^3 \cdot 6 t^2 u}
\][/tex]
### Step 5: Multiply the Constants
First, let's simplify the numerical constants. Multiply the constants in the numerator and denominator:
[tex]\[
\frac{3 \cdot 7 \cdot r \cdot t^5 \cdot u^3 \cdot s^2}{28 \cdot 6 \cdot t^2 \cdot u \cdot s^3}
\][/tex]
### Step 6: Simplify the Constants
Calculate the product of the constants:
[tex]\[
\frac{21 r t^5 u^3 s^2}{168 t^2 u s^3}
\][/tex]
### Step 7: Simplify the Fraction
We simplify the fraction by dividing the numerator and the denominator by the same factor. Here, we can divide both the numerator and the denominator by 21:
[tex]\[
\frac{r t^5 u^3 s^2}{8 t^2 u s^3}
\][/tex]
### Step 8: Simplify the Variables
Finally, simplify the variables. Begin by canceling the similar terms in the numerator and the denominator:
- [tex]\( t^5 \div t^2 = t^{5-2} = t^3 \)[/tex]
- [tex]\( u^3 \div u = u^{3-1} = u^2 \)[/tex]
- [tex]\( s^2 \div s^3 = s^{2-3} = s^{-1} = \frac{1}{s} \)[/tex]
Putting it all together, we get:
[tex]\[
\frac{r t^3 u^2}{8 s}
\][/tex]
Hence, the simplified form of the complex fraction [tex]\( \frac{\frac{3 t^5 u^3}{28 s^3}}{\frac{6 t^2 u}{7 r s^2}} \)[/tex] is:
[tex]\[
\frac{r t^3 u^2}{8 s}
\][/tex]
