Certainly! Let's carefully multiply and simplify the given rational expressions step-by-step.
We start with the given rational expressions:
[tex]\[
\frac{15ax}{2x^3} \cdot \frac{4x}{5a}
\][/tex]
### Step 1: Multiply the Numerators
First, we multiply the numerators of both fractions:
[tex]\[
15ax \cdot 4x = 15 \cdot a \cdot x \cdot 4 \cdot x = 60ax^2
\][/tex]
### Step 2: Multiply the Denominators
Next, we multiply the denominators of both fractions:
[tex]\[
2x^3 \cdot 5a = 2 \cdot x^3 \cdot 5 \cdot a = 10ax^3
\][/tex]
So we can combine the results to form a single fraction:
[tex]\[
\frac{60ax^2}{10ax^3}
\][/tex]
### Step 3: Simplify the Fraction
Now, let's simplify the fraction by cancelling common factors in the numerator and denominator.
1. Cancel common coefficient factors:
Both 60 and 10 can be divided by 10:
[tex]\[
\frac{60 \div 10}{10 \div 10} = \frac{6}{1} = 6
\][/tex]
2. Cancel common factors involving [tex]\(a\)[/tex]:
The [tex]\( a \)[/tex] in the numerator and denominator cancel each other out:
[tex]\[
\frac{a}{a} = 1
\][/tex]
3. Cancel common factors involving [tex]\(x\)[/tex]:
We have [tex]\(x^2\)[/tex] in the numerator and [tex]\(x^3\)[/tex] in the denominator. Cancel [tex]\(x^2\)[/tex] in both, which leaves [tex]\(x\)[/tex] in the denominator because:
[tex]\[
\frac{x^2}{x^3} = \frac{1}{x} = \frac{1}{x^{3-2}} = \frac{1}{x}
\][/tex]
Thus, we are left with:
[tex]\[
\frac{6}{x}
\][/tex]
### Final Answer
The simplified form of the given rational expression is:
[tex]\[
\boxed{\frac{6}{x}}
\][/tex]
