Factorize the Expression: xy/8 + xy²/16 + xy³/20 Step-by-Step

Decompose the following expression into factors:

$\frac{xy}{8}+\frac{xy^2}{16}+\frac{xy^3}{20}$

To factor the given expression, we proceed as follows:

• Step 1: Identify Common Factors
  The terms $\frac{xy}{8}$, $\frac{xy^2}{16}$, and $\frac{xy^3}{20}$ have $xy$ as a common factor in the numerators.

• Step 2: Factor Out Common Numerator
  Factor $xy$ out of each term:
  $\frac{xy}{8} = \frac{xy}{8}$,
  $\frac{xy^2}{16} = \frac{xy \cdot y}{16}$,
  $\frac{xy^3}{20} = \frac{xy \cdot y^2}{20}$.

• Step 3: Simplify the Expression
  Factor out $xy$ and adjust fractions:
  $xy \left( \frac{1}{8} + \frac{y}{16} + \frac{y^2}{20} \right)$.

• Step 4: Simplify Denominator Terms
  Factor $\frac{1}{4}$ common to $\frac{1}{8}$, $\frac{y}{16}$, and $\frac{y^2}{20}$:
  $\frac{xy}{4} \left( \frac{1}{2} + \frac{y}{4} + \frac{y^2}{5} \right)$.

Thus, the expression $\frac{xy}{8} + \frac{xy^2}{16} + \frac{xy^3}{20}$ decomposes to the factored form $\frac{xy}{4} \left( \frac{1}{2} + \frac{y}{4} + \frac{y^2}{5} \right)$.