$\frac{24a^7{b}^{14}}{24a^3{b}^6}=\frac{24}{8}\times \frac{a^7}{a^3}\times \frac{b^{14}}{b^6}=3a^{7-3}{b}^{14-6}$ $= 3a^4{b}^8$ (Ans.)

$\frac{- 78x^{17}{y}^{14}{z}^{11}}{- 13x^9{y}^5{z}^3}=\frac{-78}{- 13}\times \frac{x^{17}}{x^9}\times \frac{y^{14}}{y^5}\times \frac{z^{11}}{z^3}$

$= 6 \times x^{17-9 }\times y^{14-5} \times z^{11-3}$

$= 6x^8{y}^9{z}^8$ (Ans.)

$\frac{\left(-108a^{18}{b}^6{c}^{13}\right)}{\left(9a^{13}{b}^6{c}^9\right)}=\frac{-108}{9}\times \frac{a^{18}}{a^{13}}\times \frac{b^6}{b^6}\times \frac{c^{13}}{c^9}$

= $-12\times a^{18-13}\times b^{6-6}\times c^{13-9}$

$=-12a^5\times b^0\times c^4$

$=-12a^5{c}^4$ (Ans.)

$\frac{18x^7{y}^5- 42x^9{y}^6}{3x^4{y}^3}= \frac{18x^7{y}^5}{3x^4{y}^3}-\frac{42x^9{y}^6}{3x^4{y}^3}$

$= 6x^{7-4}{y}^{5-3}-14x^{9-4}{y}^{6-3}$

$= 6x^3{y}^2- 14x^5{y}^3$ (Ans.)

$\frac{a^6{b}^8{c}^{10} - a^9{b}^7{c}^4 - a^{12}{b}^5{c}^{11}}{a^6{b}^5{c}^4}$

$= \frac{a^6{b}^8{c}^{10}}{a^6{b}^5{c}^4}-\frac{ a^9{b}^7{c}^4}{a^6{b}^5{c}^4}-\frac{a^{12}{b}^5{c}^{11}}{a^6{b}^5{c}^4}$

= $a^{6-6}{b}^{8-5}{c}^{10-4}.-a^{9-6}{b}^{7-5}{c}^{4-4} .-a^{12-6}{b}^{5-5}{c}^{11-4}$

= $a^0.b^3.c^6 -a^3{b}^2.c^0 - a^6.b^0.c^7$

= $b^3{c}^6 - a^3{b}^{2 }- a^6{c}^7$ $\left[\because a^0= b^0= c^0 = 1\right]$(Ans.)

$\frac{- 54x^5{y}^{14} + 66x^8{y}^{15}}{- 6x^3{y}^4}$

$= \frac{- 54x^5{y}^{14}}{- 6x^3{y}^4}+\frac{66x^8{y}^{15}}{- 6x^3{y}^4}$

$= 9x^{5-3}{y}^{14-4}-11x^{8-3}{y}^{15-4}$

$= 9x^2{y}^{10} - 11x^5{y}^{11}$  (Ans.)