To solve the problem, we need to divide the two fractions [tex]\(\frac{10 x y^2}{3 z}\)[/tex] and [tex]\(\frac{5 x y}{6 z^3}\)[/tex]. Let's follow a step-by-step approach:
1. Write the division as a multiplication by the reciprocal:
[tex]\[
\frac{10 x y^2}{3 z} \div \frac{5 x y}{6 z^3} = \frac{10 x y^2}{3 z} \times \frac{6 z^3}{5 x y}
\][/tex]
2. Multiply the numerators and the denominators:
[tex]\[
= \frac{10 x y^2 \cdot 6 z^3}{3 z \cdot 5 x y}
\][/tex]
3. Simplify the expression by canceling out common factors:
- [tex]\(10 \times 6 = 60\)[/tex]
- [tex]\(3 \times 5 = 15\)[/tex]
- [tex]\(x\)[/tex] cancels out with [tex]\(x\)[/tex]
- [tex]\(y^2 \times y\)[/tex] becomes [tex]\(y\)[/tex]
- [tex]\(z^3 \div z\)[/tex] becomes [tex]\(z^2\)[/tex]
So, we have:
[tex]\[
\frac{60 y z^2}{15}
\][/tex]
4. Simplify the fraction:
- Divide the numerator and the denominator by their greatest common divisor, which is 15.
[tex]\[
\frac{60 y z^2}{15} = \frac{60}{15} \cdot y z^2 = 4 y z^2
\][/tex]
Therefore, the simplified form of the quotient is:
[tex]\[
\boxed{4 y z^2}
\][/tex]
The correct answer is C. [tex]\(4 y z^2\)[/tex].
