Question: 3. Simplify $\frac{12xy^5}{36x^9y^3}$

Solution

To simplify the expression \[ \frac{12xy^5}{36x^9y^3} \] First, simplify the numerical coefficients: \[ \frac{12}{36} = \frac{1}{3} \] Next, simplify the powers of \(x\) using the quotient rule for exponents \(\frac{x^a}{x^b} = x^{a-b}\): \[ \frac{x}{x^9} = x^{1-9} = x^{-8} \] Then, simplify the powers of \(y\) using the quotient rule: \[ \frac{y^5}{y^3} = y^{5-3} = y^2 \] Combine all parts to get the simplified expression: \[ \frac{1}{3} \cdot x^{-8} \cdot y^2 = \frac{y^2}{3x^8} \] The simplified expression is: \[ \frac{y^2}{3x^8} \]