Question

① \\(\frac{10x^{2}y^{3}}{35xy^{4}}\\) ② \\(\frac{102x^{3}y^{2}}{-18x^{2}y^{3}z}\\) ③ \\(\frac{-3x^{2}y}{15y}\\) ④ \\(\frac{4xy^{2}}{32yz}\\) ⑤ \\(\frac{-4m^{8}n}{-2m^{15}n^{7}}\\) ⑥ \\(\frac{32x^{8}y^{2}}{16x^{5}y^{3}}\\) ⑦ \\(\frac{9x^{9}y^{4}}{-27xy^{4}}\\) ⑧ \\(\frac{m^{8}n^{3}}{m^{10}n^{5}}\\)

Explanation:

Step1: Simplify coefficients and variables

For $\frac{10x^2y^3}{35xy^4}$:
Simplify coefficient: $\frac{10}{35}=\frac{2}{7}$
Apply exponent rule $\frac{x^a}{x^b}=x^{a-b}$:
$\frac{x^2}{x}=x^{2-1}=x$, $\frac{y^3}{y^4}=y^{3-4}=y^{-1}=\frac{1}{y}$
Combine: $\frac{2x}{7y}$

Step2: Simplify coefficients and variables

For $\frac{102x^3y^2}{-18x^2y^3z}$:
Simplify coefficient: $\frac{102}{-18}=-\frac{17}{3}$
Apply exponent rule:
$\frac{x^3}{x^2}=x^{3-2}=x$, $\frac{y^2}{y^3}=y^{2-3}=y^{-1}=\frac{1}{y}$
Combine: $-\frac{17x}{3yz}$

Step3: Simplify coefficients and variables

For $\frac{-3x^2y}{15y}$:
Simplify coefficient: $\frac{-3}{15}=-\frac{1}{5}$
Apply exponent rule: $\frac{y}{y}=1$, $\frac{x^2}{1}=x^2$
Combine: $-\frac{x^2}{5}$

Step4: Simplify coefficients and variables

For $\frac{4xy^2}{32yz}$:
Simplify coefficient: $\frac{4}{32}=\frac{1}{8}$
Apply exponent rule:
$\frac{y^2}{y}=y^{2-1}=y$
Combine: $\frac{xy}{8z}$

Step5: Simplify coefficients and variables

For $\frac{-4m^8n}{-2m^{15}n^7}$:
Simplify coefficient: $\frac{-4}{-2}=2$
Apply exponent rule:
$\frac{m^8}{m^{15}}=m^{8-15}=m^{-7}=\frac{1}{m^7}$, $\frac{n}{n^7}=n^{1-7}=n^{-6}=\frac{1}{n^6}$
Combine: $\frac{2}{m^7n^6}$

Step6: Simplify coefficients and variables

For $\frac{32x^8y^2}{16x^5y^3}$:
Simplify coefficient: $\frac{32}{16}=2$
Apply exponent rule:
$\frac{x^8}{x^5}=x^{8-5}=x^3$, $\frac{y^2}{y^3}=y^{2-3}=y^{-1}=\frac{1}{y}$
Combine: $\frac{2x^3}{y}$

Step7: Simplify coefficients and variables

For $\frac{9x^9y^4}{-27xy^4}$:
Simplify coefficient: $\frac{9}{-27}=-\frac{1}{3}$
Apply exponent rule:
$\frac{x^9}{x}=x^{9-1}=x^8$, $\frac{y^4}{y^4}=1$
Combine: $-\frac{x^8}{3}$

Step8: Simplify coefficients and variables

For $\frac{m^8n^3}{m^{10}n^5}$:
Apply exponent rule:
$\frac{m^8}{m^{10}}=m^{8-10}=m^{-2}=\frac{1}{m^2}$, $\frac{n^3}{n^5}=n^{3-5}=n^{-2}=\frac{1}{n^2}$
Combine: $\frac{1}{m^2n^2}$

Answer:

1. $\frac{2x}{7y}$
2. $-\frac{17x}{3yz}$
3. $-\frac{x^2}{5}$
4. $\frac{xy}{8z}$
5. $\frac{2}{m^7n^6}$
6. $\frac{2x^3}{y}$
7. $-\frac{x^8}{3}$
8. $\frac{1}{m^2n^2}$