Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \frac{4 x^{3} y^{3}}{2 x y} \cdot \frac{5 x y^{2}}{2 y} $$

Short answer

The simplified expression with no negative exponents is \(5x^{2}y^{3}\).

Step-by-step solution

Simplify within fractions

Start by multiplying the numerators and denominators of each complex fraction. Commence with \(4 \cdot x^{3} \cdot 5 \cdot x \cdot y^{2}\) for the numerator and \(2 \cdot x \cdot y \cdot 2 \cdot y\) for the denominator.

Use exponential laws

Streamlining the various exponents will help simplifying the complex fraction. Apply the following rules: If they have the same base, add the exponents for multiplication \(a^n \cdot a^m = a^{n+m}\), and subtract the exponents for the division \(a^n / a^m = a^{n-m}\). Thus, the simplified expression becomes \(20x^{4}y^{5} / 4x^{2}y^{2}\). Continuing with the simplification results in \(5x^{2}y^{3}\).

Check results

Finally, double-check the results to be sure they are free of negative exponents and as simplified as possible, which in this case, is \(5x^{2}y^{3}\).