Question

Simplify the expression. The simplified expression should have no negative exponents. $$\frac{5 x^{4} y}{3 x y^{2}} \cdot \frac{9 x y}{x^{2} y}$$

Short answer

The simplified expression is \( 15 x^{2} / y \).

Step-by-step solution

Simplify the numerators and the denominators

First, simplify the numerators and the denominators individually. In the numerator, we have \( \frac{5 x^{4} y}{3 x y^{2}} \). Now simplify this by cancelling out the common terms in the numerator and denominator.So: \( \frac{5 x^{4} y}{3 x y^{2}} = \frac{5 x^{3}}{3 y} \). Similarly, simplify the second fraction by cancelling out the common terms in the numerator and denominator.So: \( \frac{9 x y}{x^{2} y} = \frac{9}{x} \).

Multiply the simplified fractions

Now, multiply the simplified results of both fractions. So, \( \frac{5 x^{3}}{3 y} \cdot \frac{9}{x} =\frac{5*9 x^{3}}{3*y*x} = 15 x^{2} / y \). This is the simplified form of the expression after multiplying the simplified fractions. Note that we have reduced the exponents by subtracting them. When terms with the same base are divided, we subtract the exponents.

Check for negative exponents

Finally, check if the simplified expression has any negative exponents. If it has, apply the rule for negative exponents which states that a^(-n) = 1 / a^n. But here, in the solution \( 15 x^{2} / y \), there are no negative exponents, hence no further simplification is necessary.