To solve the problem of multiplying the rational expressions and simplifying, let's break it down step by step:
Given:
[tex]\[
\frac{8 t^2}{5 s} \cdot \frac{15 s}{4 t}
\][/tex]
### Step 1: Multiply the numerators
First, we multiply the numerators of the two fractions:
[tex]\[
8 t^2 \cdot 15 s = 8 \cdot 15 \cdot t^2 \cdot s = 120 t^2 s
\][/tex]
### Step 2: Multiply the denominators
Next, we multiply the denominators of the two fractions:
[tex]\[
5 s \cdot 4 t = 5 \cdot 4 \cdot s \cdot t = 20 st
\][/tex]
So, the product of the rational expressions before simplification is:
[tex]\[
\frac{120 t^2 s}{20 st}
\][/tex]
### Step 3: Simplify the expression
To simplify the fraction, we need to cancel out common factors in the numerator and the denominator.
1. Numerical coefficients: The numerical part of the fraction is [tex]\(\frac{120}{20}\)[/tex]:
[tex]\[
\frac{120}{20} = 6
\][/tex]
2. Variable [tex]\(t\)[/tex]: In the numerator, we have [tex]\(t^2\)[/tex] and in the denominator, we have [tex]\(t\)[/tex]. So, [tex]\(t^2 / t = t\)[/tex]:
[tex]\[
t^2 / t = t
\][/tex]
3. Variable [tex]\(s\)[/tex]: In both the numerator and the denominator, we have [tex]\(s\)[/tex], which cancels out:
[tex]\[
s / s = 1
\][/tex]
So, the simplified form of the product is:
[tex]\[
\frac{120 t^2 s}{20 st} = 6t
\][/tex]
### Conclusion
Thus, the simplified form of the given multiplication of rational expressions [tex]\(\frac{8 t^2}{5 s} \cdot \frac{15 s}{4 t}\)[/tex] is:
[tex]\[
6t
\][/tex]
We have carefully multiplied and simplified the expressions step by step to arrive at this result.
