Certainly! Let's simplify and find the product of the given expressions step by step.
We need to find the product of:
[tex]\[
\frac{48 x^5 y^3}{y^4} \cdot \frac{x^2 y}{6 x^3 y^2}
\][/tex]
1. Simplify the first expression:
[tex]\[
\frac{48 x^5 y^3}{y^4}
\][/tex]
Since \( \frac{y^3}{y^4} = y^{3-4} = y^{-1} \), the expression simplifies to:
[tex]\[
\frac{48 x^5 y^3}{y^4} = 48 x^5 \cdot y^{-1} = \frac{48 x^5}{y}
\][/tex]
2. Simplify the second expression:
[tex]\[
\frac{x^2 y}{6 x^3 y^2}
\][/tex]
We can simplify the \( x \) and \( y \) terms individually:
For \( x \): \( \frac{x^2}{x^3} = x^{2-3} = x^{-1} \)
For \( y \): \( \frac{y}{y^2} = y^{1-2} = y^{-1} \)
Therefore, the expression simplifies to:
[tex]\[
\frac{x^2 y}{6 x^3 y^2} = \frac{x^{-1} y^{-1}}{6} = \frac{1}{6 x y}
\][/tex]
3. Multiply the simplified expressions:
Now we multiply the two simplified expressions:
[tex]\[
\left(\frac{48 x^5}{y}\right) \cdot \left(\frac{1}{6 x y}\right)
\][/tex]
Simplify the multiplication by multiplying numerators and denominators:
[tex]\[
\frac{48 x^5}{y} \cdot \frac{1}{6 x y} = \frac{48 x^5 \cdot 1}{y \cdot 6 x y} = \frac{48 x^5}{6 x y^2}
\][/tex]
Next, simplify the fraction \( \frac{48}{6} \):
[tex]\[
\frac{48}{6} = 8
\][/tex]
For the \( x \) terms: \( \frac{x^5}{x} = x^{5-1} = x^4 \)
Hence, the product becomes:
[tex]\[
\frac{8 x^4}{y^2}
\][/tex]
Therefore, the product of the given expressions is:
[tex]\[
\boxed{\frac{8 x^4}{y^2}}
\][/tex]
