Tag: composition of ratios

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 $\sqrt {9} : \sqrt {16} = 3 : 4$