Find the multiplicative inverse (or reciprocal) of each of the following rational numbers :

(i) $\dfrac{6}{25}$

(ii) $-2\dfrac{3}{11}$

(iii) $\dfrac{-8}{9}$

(iv) $\dfrac{-23}{16}$

(v) 12

(vi) $\dfrac{1}{10}$

(vii) -6

(viii) -1

(ix) $\dfrac{-1}{5}$

(x) $\dfrac{-7}{-9}$

(i) $\dfrac{6}{25}$

Since, $\dfrac{6}{25}$ x $\dfrac{25}{6}$ = 1

∴ The multiplicative inverse of $\dfrac{6}{25}$ is $\dfrac{25}{6}$.

(ii) $-2\dfrac{3}{11}$

First, convert the mixed fraction to an improper fraction: $\dfrac{-(2 \times 11 + 3)}{11} = \dfrac{-25}{11}$.

Since, $\dfrac{-25}{11}$ x $\dfrac{-11}{25}$ = 1

∴ The multiplicative inverse of $\dfrac{-25}{11}$ is $\dfrac{-11}{25}$.

(iii) $\dfrac{-8}{9}$

Since, $\dfrac{-8}{9}$ x $\dfrac{-9}{8}$ = 1

∴ The multiplicative inverse of $\dfrac{-8}{9}$ is $\dfrac{-9}{8}$.

(iv) $\dfrac{-23}{16}$

Since, $\dfrac{-23}{16}$ x $\dfrac{-16}{23}$ = 1

∴ The multiplicative inverse of $\dfrac{-23}{16}$ is $\dfrac{-16}{23}$.

(v) 12

Express 12 as $\dfrac{12}{1}$

Since, $\dfrac{12}{1}$ x $\dfrac{1}{12}$ = 1

∴ The multiplicative inverse of $\dfrac{12}{1}$ is $\dfrac{1}{12}$.

(vi) $\dfrac{1}{10}$

Since, $\dfrac{1}{10}$ x $\dfrac{10}{1}$ = 1

∴ The multiplicative inverse of $\dfrac{1}{10}$ is $\dfrac{10}{1}$, which is 10.

(vii) -6

Express -6 as $\dfrac{-6}{1}$.

Since, $\dfrac{-6}{1}$ x $\dfrac{-1}{6}$ = 1

∴ The multiplicative inverse of $\dfrac{-6}{1}$ is $\dfrac{-1}{6}$.

(viii) -1

Express -1 as $\dfrac{-1}{1}$.

Since, $\dfrac{-1}{1}$ x $\dfrac{-1}{1}$ = 1

∴ The multiplicative inverse of $\dfrac{-1}{1}$ is $\dfrac{-1}{1}$, which is still -1.

(ix) $\dfrac{-1}{5}$

Since, $\dfrac{-1}{5}$ x $\dfrac{-5}{1}$ = 1

∴ The multiplicative inverse of $\dfrac{-1}{5}$ is $\dfrac{-5}{1}$, which is -5.

(x) $\dfrac{-7}{-9}$

First, simplify the fraction: $\dfrac{-7}{-9} = \dfrac{7}{9}$.

Since, $\dfrac{7}{9}$ x $\dfrac{9}{7}$ = 1

∴ The multiplicative inverse of $\dfrac{7}{9}$ is $\dfrac{9}{7}$.