Question

algebra: concepts and connections - plc multiplying and dividing rational expressions what is the missing factor in the equation? assume ( x
eq 0 ) and ( y
eq 0 ) ( left( \frac{6x^2}{5y}
ight) (?) = \frac{3xy^2}{10} ) options: ( \frac{9x^3y}{25} ), ( \frac{2xy^2}{5} ), ( \frac{3x}{5y} ), ( \frac{y^2}{4x} )

Explanation:

Step1: Isolate the missing factor

Let the missing factor be $M$. Rearrange the equation to solve for $M$:
$$M = \frac{\frac{3xy^2}{10}}{\frac{6x^2}{5y}}$$

Step2: Rewrite as multiplication by reciprocal

Dividing by a fraction equals multiplying by its reciprocal:
$$M = \frac{3xy^2}{10} \times \frac{5y}{6x^2}$$

Step3: Multiply numerators and denominators

Combine the fractions by multiplying across:
$$M = \frac{3xy^2 \times 5y}{10 \times 6x^2}$$

Step4: Simplify coefficients and variables

Simplify constants ($\frac{3 \times 5}{10 \times 6} = \frac{15}{60} = \frac{1}{4}$) and variables ($\frac{x y^3}{x^2} = \frac{y^3}{x}$):
$$M = \frac{y^3}{4x}$$

Answer:

$\frac{y^3}{4x}$