Question

(\frac{12x y^{5}}{15x^{5}y^{3}})

Explanation:

Step1: Simplify coefficients

Simplify the fraction of the coefficients \( \frac{12}{15} \). We can divide both numerator and denominator by their greatest common divisor, which is 3. So \( \frac{12\div3}{15\div3}=\frac{4}{5} \).

Step2: Simplify \( x \)-terms

For the \( x \)-terms, we use the rule of exponents \( \frac{x^m}{x^n}=x^{m - n} \). Here, \( m = 1 \) (since \( x=x^1 \)) and \( n = 5 \), so \( \frac{x}{x^5}=x^{1-5}=x^{-4}=\frac{1}{x^4} \) (using the rule \( a^{-n}=\frac{1}{a^n} \)).

Step3: Simplify \( y \)-terms

For the \( y \)-terms, using the same exponent rule \( \frac{y^m}{y^n}=y^{m - n} \), where \( m = 5 \) and \( n = 3 \), so \( \frac{y^5}{y^3}=y^{5 - 3}=y^2 \).

Step4: Combine all simplified terms

Multiply the simplified coefficient, \( x \)-term, and \( y \)-term together: \( \frac{4}{5}\times\frac{1}{x^4}\times y^2=\frac{4y^2}{5x^4} \).

Answer:

\( \frac{4y^2}{5x^4} \)