Solve the Fraction Puzzle: Simplify 85x×y³/5y⁴x³ × 9xy/3yx²

To solve this problem, we'll simplify each part of the given expression step by step:

Original expression:
$\frac{85x \cdot y^3}{5y^4x^3} \cdot \frac{9xy}{3yx^2}$.

Let's simplify the first fraction:
$\frac{85x \cdot y^3}{5y^4x^3}$.

• Divide the coefficients: $\frac{85}{5} = 17$.
• Apply the quotient rule of exponents:
  $\frac{x}{x^3} = x^{1-3} = x^{-2}$ and $\frac{y^3}{y^4} = y^{3-4} = y^{-1}$.
• This simplifies to: $17x^{-2}y^{-1}$.

Now, simplify the second fraction:
$\frac{9xy}{3yx^2}$.

• Divide the coefficients: $\frac{9}{3} = 3$.
• Cancel the $y$: $\frac{y}{y} = 1$.
• Apply the quotient rule of exponents: $\frac{x}{x^2} = x^{1-2} = x^{-1}$.
• This simplifies to: $3x^{-1}$.

Multiply the simplified expressions:
$(17x^{-2}y^{-1})\cdot (3x^{-1})$.

• Combine the coefficients: $17 \cdot 3 = 51$.
• Apply the product rule for $x$: $x^{-2} \cdot x^{-1} = x^{-2-1} = x^{-3}$.
• For $y$, simply note: $y^{-1}$.

Thus, the simplified expression is: $51x^{-3}y^{-1}$.