The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b} = (a)^{\frac {1}{3}} : (b)^{\frac {1}{3}}$

$ \dfrac{y}{x-z}=\dfrac{y+x}{z}=\dfrac{x}{y} $

Now,

$ \dfrac{y}{x-z}=\dfrac{x}{y} $

$ {{y}^{2}}={{x}^{2}}-xz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(1) $

And

$ \dfrac{y+x}{z}=\dfrac{x}{y} $

$ {{y}^{2}}+xy=xz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(2) $

$ {{x}^{2}}-xz+xy=xz $

$ x-z+y=z $

$ 2z=x+y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(3) $

$ And $

$ \dfrac{y}{x-z}=\dfrac{y+x}{z} $

$ yz=xy-yz+{{x}^{2}}-xz $

$ 2yz=xy+{{x}^{2}}-xz $

$ 2yz=x\left( y+x \right)-xz $                    [From equation (3)]

$ 2yz=2xz-xz $

$ 2yz=xz $

$ 2y=x $

$ \dfrac{x}{y}=\dfrac{2}{1}\,\,\,\,\,\,\,\,......\,\,\left( 4 \right) $

Substituting this value in equation (3), we get

$ 2z=2y+y $

$ 2z=3y $

$ \dfrac{y}{z}=\dfrac{2}{3}\,\,\,\,\,......\,\,\left( 5 \right) $

By equation (4) and (5), we get

$ x:y:z=4:2:3 .$

Hence, this is the answer.

Given,

Given, $\displaystyle \frac{a}{2} = \frac{b}{3} = \frac{c}{4}$

Given $a:b=3:4$

By the defination of compound ratio these ratio can be expressed as
$\dfrac {x}{y} \times \dfrac {y}{z}\times \dfrac {z}{w} = \dfrac {x}{w}$
Hence $x : w$

$\dfrac{a}{b} = \dfrac{5}{7} $       $\dfrac{b}{c} = \dfrac{6}{11}$

Required compounded ratio $\displaystyle=\frac{2}{3}\times\frac{6}{11}\times\frac{11}{2}=2:1$

By the definition of compounded ratio these ratio can be expressed as
$\dfrac {(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \dfrac {(x^{4} - y^{4})}{(x + y)^{4}}$
$\dfrac {(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \dfrac {(x^{2} - y^{2})(x^{2} + y^{2})}{(x + y)^{4}}$
$= \dfrac {(x^{2} - y^{2})^{2}}{(x + y)^{4}}$
$= \dfrac {[(x - y) (x + y)]^{2}}{(x + y)^{4}}$
$= \dfrac {(x - y)^{2}}{(x + y)^{2}}$
Hence $(x - y)^{2} : (x + y)^{2}$

Given, 

Given, $\displaystyle \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = 3 : 4 : 5$

By the defination of compound ratio $4 : 5, 10 : 6$ and $3 : 5$ can be expressed as
$\dfrac {4}{5}\times \dfrac {10}{6} \times \dfrac {3}{5} = \dfrac {4}{5}$
Hence $4 : 5$

By the defination of compound ratio these ratio can be expressed as
$\dfrac {3}{5}\times \dfrac {7}{9}\times \dfrac {15}{28} = \dfrac {1}{4}$
Hence $1 : 4$

By the defination of compound ratio $5 : 7, 12 : 8$ and $13 : 6$ can be expressed as
$\dfrac {5}{7}\times \dfrac {12}{8}\times \dfrac {13}{6} = \dfrac {65}{28}$
Hence $65 : 28$

By the defination of compound ratio $9 : 27$ and $4 : 12$ can be expressed as
$\dfrac {9}{27} \times \dfrac {4}{12} = \dfrac {1}{9}$
Hence $1 : 9$

let be $ a=\dfrac{3}{5}, b=\dfrac{7}{8}, c=\dfrac{5}{10}$
therefore the $ratio=a\times b\times c=\dfrac{3\times 7\times 5}{5\times 8\times 10}=\dfrac{21}{80}$

By the defination of compound ratio $5 : 7, 14 : 10$ and $2 : 3$ can be expressed as
$\dfrac {5}{7} \times \dfrac {14}{10}\times \dfrac {2}{3} = \dfrac {2}{3}$
Hence $2 : 3$

By the defination of compound ratio $l : m, m : n$ and $n : o$ can be expressed as
$\dfrac {l}{m}\times \dfrac {m}{n} \times \dfrac {n}{o} = \dfrac {l}{o}$
Hence $l : o$

By the defination of compound ratio $18 : 9, 16 : 13$ and $6 : 9$ can be expressed as
$\dfrac {18}{9}\times \dfrac {16}{13}\times \dfrac {6}{9} = \dfrac {64}{39}$
Hence $64 : 39$

By the defination of compound ratio $2 : 3$ and $5 : 7$ can be expressed as
$\dfrac {2}{3}\times \dfrac {5}{7} = \dfrac {10}{21}$
Hence $10 : 21$

By the defination of compound ratio $10 : 30$ and $60 : 80$
$\dfrac {10}{30}\times \dfrac {60}{80} = \dfrac {1}{4}$
Hence $1 : 4$.

The duplicate ratio of $a : b$ is $b : a$
$\therefore$ The duplicate ratio of $3a : 4b$ is $(3a)^{2} : (4b)^{2} = 9a^{2} : 16b^{2}$

The duplicate ratio of $a : b$ is $b : a$
$\therefore$ The duplicate ratio of $\sqrt {2} : \sqrt {3}$ is $(\sqrt {2})^{2} : (\sqrt {3})^{2} = 2 : 3$

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$.
$\therefore$ The duplicate ratio of $5 : 7$ is $5^{2} : 7^{2} = 25 : 49$

The duplicate ratio of $a : b$ is $b : a$
$\therefore$ The duplicate ratio of $\dfrac {x}{2} : \dfrac {y}{3}$ is $\left (\dfrac {x}{2}\right )^{2} : \left (\dfrac {y}{3}\right )^{2} = \dfrac {x^{2}}{4} : \dfrac {y^{2}}{9}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $2 : 5$ is $2^{3} : 5^{3} = 8 : 125$.

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$
$\therefore$ The duplicate ratio of $2\sqrt {2} : 3\sqrt {5}$ is $(2\sqrt {2})^{2} : (3\sqrt {5})^{2} = 8 : 45$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $a^{3} : b^{3}$ is $(a^{3})^{3} : (b^{3})^{3} = a^{9} : b^{9}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $4 : 7$ is $4^{3} : 7^{3} = 64 : 243$

By the defination of compound ratio these ratio can be expressed as
$\dfrac {a - b}{a + b}\times \dfrac {(a - b)^{2}}{(a + b)^{2}} \times \dfrac {(a + b)^{3}}{(a - b)^{3}} = \dfrac {1}{1}$
Hence, $1 : 1$.

$\sqrt [3]{9} : \sqrt {8} = (9)^{\dfrac {1}{3}} : (8)^{\dfrac {1}{2}}$
The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\sqrt [3]{9} : \sqrt {8}$ is $[(9)^{\dfrac {1}{3}}]^{3} : [(8)^{\dfrac {1}{2}}]^{3}$
$= 9 : (2\sqrt {2})^{3}$
$= 9 : 8(\sqrt {2})^{3}$
$= 9 : 16\sqrt {2}$.

The subduplicate ratio of a:ba:b

The sub-duplicate ratio of $x : y$ is $\sqrt {x} : \sqrt {y}$
$\therefore$ The subduplicate ratio of $9a^{2}b^{2} : 16r^{2}s^{2}$ is $\sqrt {9a^{2}b^{2}} : \sqrt {16r^{2}s^{2}}$
$= 3ab : 4rs$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $a^{4} : b^{4}$ is $\sqrt {a^{4}} : \sqrt {b^{4}} = a^{2} : b^{2}$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $\dfrac {1}{25} : \dfrac {1}{64}$ is $\sqrt {\dfrac {1}{25}} : \sqrt {\dfrac {1}{64}} = \dfrac {1}{5} : \dfrac {1}{8}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\sqrt [3]{b^{2}} : \sqrt [3]{a^{2}}$ is $(\sqrt [3]{b^{2}})^{3} : (\sqrt [3]{a^{2}})^{3} = b^{2} : a^{2}$.

The sub-duplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$.
$\therefore$ The subduplicate ratio of $9 : 16$ is $\sqrt {9} : \sqrt {16} = 3 : 4$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $a^{2} : b^{2}$ is $(a^{2})^{3} : (b^{2})^{3} = a^{6} : b^{6}$.

The triplicate ratio of $ab^{\frac {2}{3}} : a^{\frac {2}{3}} b$ is $(ab^{\frac {2}{3}})^{3} : (a^{\frac {2}{3}}b)^{3}$
$= a^{3} \cdot b^{2} : a^{2}b^{3}$
$a^{3}b^{2} : a^{2}b^{3} = \dfrac {a^{3}b^{2}}{a^{2}b^{3}} = \dfrac {a}{b}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\dfrac {x}{2} : \sqrt [3]{x}$ is $\left (\dfrac {x}{2}\right )^{3} : \left (\sqrt [3]{x}\right )^{3} = \dfrac {x^{3}}{8} : x = \dfrac {x^{2}}{8}$
$\therefore x^{2} : 8$.

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $49 : 169$ is $\sqrt {49} : \sqrt {169} = 7 : 13$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ The subtriplicate ratio of $125y^{3} : 1000z^{6}$ is $\sqrt [3]{125y^{3}} : \sqrt [3]{1000z^{6}} = 5y : 10z^{2}$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ the subtriplicate ratio of $8x^{3} : y^{3}$ is $\sqrt [3]{8x^{3}} : \sqrt [3]{y^{3}} = 2x : y$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $64 m^{6} : 27 n^{3}$ is $\sqrt [3]{64m^{6}} : \sqrt [3]{27n^{3}}$
$= 4m^{2} : 3n$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $121 : 100$ is $\sqrt {121} : \sqrt {100} = 11 : 10$

The subduplicate ratio of $\dfrac {1}{4} : \dfrac {1}{36}$ is $\sqrt {\dfrac {1}{4}} : \sqrt {\dfrac {1}{36}} = \dfrac {1}{2} : \dfrac {1}{6}$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $x^{4} : y^{4}$ is $\sqrt {x^{4}} : \sqrt {y^{4}} = x^{2} : y^{2}$.

The subduplicate ratio of $x^{4} : y^{2}$ is $\sqrt {x^{4}} : \sqrt {y^{2}} = x^{2} : y$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $27 : 125$ is $\sqrt [3]{27} : \sqrt [3]{125} = 3 : 5$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $a^{3} : b^{3}$ is $\sqrt [3]{a^{3}} : \sqrt [3]{b^{3}} = a : b$.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $39 : 41$ is $41 : 39$

$a : b = 1 : 1$ and $b : c = 81 : 24$
$\therefore \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {1}{1} \times \dfrac {81}{24} = \dfrac {27}{8}$
$\therefore a : c = 27 : 8$
$\therefore$ The subtriplicate ratio of $27 : 8$ is $\sqrt [3]{27} : \sqrt [3]{8} = 3 : 2$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $4913 : 2744$ is $\sqrt [3]{4913} : \sqrt [3]{2744} = 17 : 14$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $\sqrt {x} : \sqrt {y} = x^{\cfrac {1}{2}} : y^{\frac {1}{2}}$ is $(x^{\cfrac {1}{2}})^{\cfrac {1}{3}} : (y^{\cfrac {1}{2}})^{\cfrac {1}{3}}$
$= x^{\cfrac {1}{6}} : y^{\cfrac {1}{6}}$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ The subtriplicate ratio of $64 : 729$ is $\sqrt [3]{64} = \sqrt [3]{729} = \sqrt [3]{(4)^{3}} : \sqrt [3]{(9)^{3}} = 4 : 9$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
Now, $3 : 81 = \dfrac {3}{81} = \dfrac {1}{27} = 1 : 27$
$\therefore$ The subtriplicate ratio of $1 : 27$ is $\sqrt [3]{1} : \sqrt [3]{27} = 1 : 3$

The reciprocal ratio of a:ba:b.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ The subtriplicate ratio of $216 : 1331$ is $\sqrt [3]{216} : \sqrt [3]{1331} = \sqrt [3]{(6)^{3}} : \sqrt [3]{(11)^{3}}$
$= 6 : 11$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $2 : 3$ is $\sqrt [3]{2} : \sqrt [3]{3}$.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $2 : 3$ is $3 : 2$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $9x^{2} : 11x^{2}$ is $11x^{2} : 9x^{2}$

The reciprocal ratio of a:ba:b.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $121x^{3} : 25x^{3}$ is $25x^{3} : 121x^{3}$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $81 : 121$ is $121 : 81$.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $8 : 15$ is $15 : 8$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $(x - 5) : (x - 7)$ is $(x - 7) : (x - 5)$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $(x + 12) : (x + 21)$ is $(x + 21) : (x + 12)$

we have to find the ratio compound of $2:3$ and sub-duplicate ratio of $4:9$ 

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $21 : 31$ is $31 : 21$

$(4x + 7y) : (5x - y)$ is the duplicate ratio of $5 : 1$.
Also, the duplicate ratio of $5 : 1$ is $25 : 1$.
$\therefore \dfrac {4x + 7y}{5x - y} = \dfrac {25}{1}\Rightarrow 4x + 7y = 125x - 25y$
$\therefore 125x - 4x = 7y + 25y$
$\therefore 121x = 32y$
$\therefore \dfrac {x}{y} = \dfrac {32}{121}$
$\therefore x : y = 32 : 121$

The duplicate ratio of $a : b$ is $a^{2}:b^{2}$
$\therefore$ The duplicate ratio of $3 : 1$ is $9 : 1$.
$\therefore \dfrac {x + y}{x - y} = \dfrac {9}{1}$
$\therefore x + y = 9x - 9y$
$\therefore y + 9y = 9x - x$
$\therefore 10y = 8x$
$\therefore \dfrac {x}{y} = \dfrac {10}{8} = \dfrac {5}{4}$
$\therefore x : y = 5 : 4$.

$(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$.
Also, the duplicate ratio of $5 : 6$ is $5^{2} : 6^{2} = 25 : 36$
$\therefore \dfrac {3x + 1}{5x - 4} = \dfrac {25}{36}$
$\therefore 108 x + 36 = 125x - 100$
$\therefore 125x - 108x = 36 + 100$
$\therefore 17x = 136$
$\therefore x = \dfrac {136}{17} = 8$

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$
$\therefore$ The duplicate ratio of $\dfrac {1}{6} : \dfrac {1}{5}$ is $\left (\dfrac {1}{6}\right )^{2} : \left (\dfrac {1}{5}\right )^{2} = \left (\dfrac {1}{36}\right ) : \left (\dfrac {1}{25}\right )$
$= \dfrac {\dfrac {1}{36}}{\dfrac {1}{25}} = \dfrac {25}{36}$.

$x : y = 4 : 9$
$y : z = 3 : 8$
$\therefore \dfrac {x}{y} \times \dfrac {y}{z} = \dfrac {4}{9}\times \dfrac {3}{8}$
$\therefore \dfrac {x}{z} = \dfrac {1}{6}\Rightarrow x : z = 1 : 6$
$\therefore$ The duplicate ratio of $x : z$ is $(1)^{2} : (6)^{2} = 1 : 36$.

$a : b = 2 : 3$
$b : c = 9 : 8$
$\therefore \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {2}{3} \times \dfrac {9}{8}$
$\therefore \dfrac {a}{c} = \dfrac {3}{4} \Rightarrow a : c = 3 : 4$.
$\therefore$ The duplicate ratio of $3 : 4$ is $3^{2} : 4^{2} = 9 : 16$

Let the salaries of Laxman and Gopal one year before be ${L} _{1} \; & \; {G} _{1}$ respectively and now be ${L} _{2} \; & \;  {G} _{2}$ respectively.

The triplicate ratio of $3 : 4$ is $3^{3} : 4^{3} = 27 : 64$
$\therefore \dfrac {4x + 3}{9x + 10} = \dfrac {27}{64}$
$\Rightarrow 256x + 192 = 243x + 270$
$\Rightarrow 256x - 243x = 270 - 192$
$\Rightarrow 13x = 78$
$\Rightarrow x = \dfrac {78}{13} = 6$
$\Rightarrow x = 6$

The triplicate ratio of $4 : 3$ is $4^{3} : 3^{3}$
$\therefore \dfrac {5x + 3}{3x + 1} = \dfrac {64}{27}$
$\Rightarrow 135x + 81 = 192x + 64$
$\Rightarrow 192x - 135x = 81 - 64$
$\Rightarrow 57x = 17$
$\therefore x = \dfrac {17}{57}$

$x : y = 2 : 5, y : z = 15 : 8, z : w = 3 : 2$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z}\times \dfrac {z}{w} = \dfrac {2}{5} \times \dfrac {15}{8}\times \dfrac {3}{2} = \dfrac {9}{8}$
$\therefore x : w = 9 : 8$
$\therefore$ The triplicate ratio of $9 : 8$ is $9^{3} : 8^{3} = 729 : 512$

The subduplicate ratio of $9 : 1$ is $\sqrt {9} : \sqrt {1} = 3 : 1$
$\therefore \dfrac {x + y}{x - y} = \dfrac {3}{1} \Rightarrow x + y = 3x - 3y$
$\Rightarrow 2x = 4y \Rightarrow \dfrac {x}{y} = \dfrac {4}{2} = \dfrac {2}{1}$

The triplicate ratio of $1 : 2$ is $1^{3} : 2^{3} = 1 : 8$
$\therefore \dfrac {x - 4}{x + 2} = \dfrac {1}{8}\Rightarrow 8x - 32 = x + 2$
$\Rightarrow 8x - x = 2 + 32$
$\Rightarrow 7x = 34$
$\Rightarrow x = \dfrac {34}{7}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $(x + y)^{\frac {2}{3}} : (x - y)^{\frac {2}{3}}$ is $[(x + y)^{\frac {2}{3}}]^{3} : [(x - y)^{\frac {2}{3}}]^{3}$
$= (x + y)^{2} : (x - y)^{2}$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $25x^{2} : 196y^{2}$ is $\sqrt {25x^{2}} : \sqrt {196y^{2}}$
$= 5x : 14y$.

The given ratio can be simplified as $\sqrt {16} : \sqrt {64} = 4 : 8 = 1 : 2$
Now, the duplicate ratio of $a : b$ is $a^{2} : b^{2}$ and so that of $1 : 2$ is $1^{2} : 2^{2} = 1 : 4$.

$\dfrac {(x^{2} - y^{2})^{2}}{(x + y)^{2}} = \dfrac {((x - y)(x + y))^{2}}{(x + y)^{2}} = \dfrac {(x - y)^{2}}{1}$
$\therefore$ The subduplicate ratio of $(x - y)^{2} : 1$ is $\sqrt {(x - y)^{2}} : \sqrt {1} = (x - y) : 1$.

$x : y = 3 : 8$ and $y : z = 4 : 9$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z} = \dfrac {3}{8} \times \dfrac {4}{9} = \dfrac {1}{6}$
$\therefore x : z = 1 : 6$
$\therefore$ The triplicate ratio of $x : z$ is $x^{3} : z^{3} = (1)^{3} : (6)^{3} = 1 : 216$

$x : y = 30 : 20$ and $y : z = 24 : 25$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z} = \dfrac {30}{20} \times \dfrac {24}{25} = \dfrac {36}{25}$
$\therefore x : z = 36 : 25$
$\therefore$ The subduplicate ratio of $x : z$ is $\sqrt {36} : \sqrt {25} = 6 : 5$

$x : y = 1 : 2$ and $y : z = 8 : 3$
$\therefore x : z = \dfrac {x}{y} \times \dfrac {y}{z} = \dfrac {1}{2} \times \dfrac {8}{3} = \dfrac {4}{3} = 4 : 3$
Now, the reciprocal ratio of $z : x$ is $x : z$
$\therefore x : z = 4 : 3$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $(x^{4} - y^{4})^{3} : (x^{2} + y^{2})^{6}$ is $\sqrt [3]{(x^{4} - y^{4})^{3}} : \sqrt [3]{(x^{2} + y^{2})^{6}} = (x^{4} - y^{4}) : (x^{2} + y^{2})^{2}$
$= \dfrac {x^{4} - y^{4}}{(x^{2} + y^{2})^{2}} = \dfrac {(x^{2} - y^{2})(x^{2} + y^{2})}{(x^{2} + y^{2})^{2}}$
$= \dfrac {x^{2} - y^{2}}{x^{2} + y^{2}} = x^{2} - y^{2} : x^{2} + y^{2}$

The reciprocal ratio of $\dfrac {1}{a} : \dfrac {1}{b}$ is $a : b$
$\therefore$ The reciprocal ratio of $\dfrac {1}{7} : \dfrac {1}{8}$ is $7 : 8$.

$a : b = 2 : 3$ and $b : c = 4 : 7$
$\therefore a : c = \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {2}{3} \times \dfrac {4}{7}$
$= \dfrac {8}{21}$
$\therefore a : c = 8 : 21$
$\therefore$ The reciprocal ratio of $a : c$ is $21 : 8$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $(a + b)^{3} : (a^{2} - b^{2})^{3}$ is $\sqrt [3]{(a + b)^{3}} : \sqrt [3]{(a^{2} - b^{2})^{3}} = (a + b) : (a^{2} - b^{2})$
$= \dfrac {a + b}{a^{2} - b^{2}} = \dfrac {a + b}{(a - b)(a + b)} = \dfrac {1}{a - b} = 1 : (a - b)$.