Question

which expression is equivalent to \\(\frac{9x^{5}y^{16}}{45x^{5}y^{4}}\\)?
\\(\frac{y^{12}}{5}\\)
\\(\frac{y^{12}}{36}\\)
\\(\frac{xy^{4}}{5}\\)
\\(\frac{xy^{4}}{36}\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients: $\frac{9}{45}=\frac{1}{5}$.

Step2: Simplify the \(x\)-terms

For the \(x\)-terms, use the rule $\frac{x^m}{x^n}=x^{m - n}$. Here, $m = 5$ and $n = 5$, so $\frac{x^5}{x^5}=x^{5 - 5}=x^0 = 1$ (since any non - zero number to the power of 0 is 1).

Step3: Simplify the \(y\)-terms

For the \(y\)-terms, use the rule $\frac{y^m}{y^n}=y^{m - n}$. Here, $m = 16$ and $n = 4$, so $\frac{y^{16}}{y^4}=y^{16 - 4}=y^{12}$.

Step4: Combine the results

Multiply the results of the coefficient, \(x\)-term, and \(y\)-term simplifications: $\frac{1}{5}\times1\times y^{12}=\frac{y^{12}}{5}$.

Answer:

$\frac{y^{12}}{5}$ (corresponding to the first option: $\boldsymbol{\frac{y^{12}}{5}}$)