(i) ${}^{13}/{}_{3125}$
(ii) ${}^{17}/{}_{8}$
(iii) ${}^{64}/{}_{455}$
(iv) ${}^{15}/{}_{1600}$
(v) ${}^{29}/{}_{343}$
(vi)$23/{{2}^{3}}{{5}^{2}}$
(vii) ${}^{129}/{}_{{{2}^{2}}{{5}^{7}}{{7}^{5}}}$
(viii) ${}^{6}/{}_{15}$
(ix) ${}^{35}/{}_{50}$
(x) ${}^{77}/{}_{210}$
Note: If the denominator has only factors of 2 and 5 or in the form of ${{2}^{m}}\times {{5}^{n}}$ then it has terminating decimal expansion.
If the denominator has factors other than 2 and 5 then we can say that it has a non-terminating decimal expansion.
(i) ${}^{13}/{}_{3125}$
Factorizing the denominator, we get,
$3125=5\times 5\times 5\times 5\times 5={{5}^{5}}$
Since, the denominator has only 5 as its factor, ${}^{13}/{}_{3125}$ has a terminating decimal expansion.
(ii)${}^{17}/{}_{8}$
Factorizing the denominator, we get,
$8=2\times 2\times 2$
Since, the denominator has only 2 as its factor, ${}^{17}/{}_{8}$ has a terminating decimal expansion.
(iii) ${}^{64}/{}_{455}$
Factorizing the denominator, we get,
$455=5\times 7\times 13$
Since, the denominator is not in the form of ${{2}^{m}}\times {{5}^{n}}$ ,thus ${}^{64}/{}_{455}$ has a non-terminating decimal expansion.
(iv)${}^{15}/{}_{1600}$
Factorizing the denominator, we get,
$1600={{2}^{6}}\times {{5}^{2}}$
Since, the denominator is in the form of ${{2}^{m}}\times {{5}^{n}}$ , thus ${}^{15}/{}_{1600}$ has a terminating decimal expansion.
(v)${}^{29}/{}_{343}$
Factorizing the denominator, we get,
$343=7\times 7\times 7$
Since, the denominator is not in the form of ${{2}^{m}}\times {{5}^{n}}$ thus ${}^{29}/{}_{343}$ has a non-terminating decimal expansion.
(vi)${}^{23}/{}_{{{2}^{3}}{{5}^{2}}}$
Clearly, the denominator is in the form of ${{2}^{m}}\times {{5}^{n}}$
Hence, ${}^{23}/{}_{{{2}^{3}}{{5}^{2}}}$ has a terminating decimal expansion.
(vii) ${}^{129}/{}_{{{2}^{2}}{{5}^{7}}{{7}^{5}}}$
As you can see, the denominator is not in the form of ${{2}^{m}}\times {{5}^{n}}$
Hence, ${}^{129}/{}_{{{2}^{2}}{{5}^{7}}{{7}^{5}}}$has a non-terminating decimal expansion.
(viii) ${}^{6}/{}_{15}$
${}^{6}/{}_{15}={}^{2}/{}_{5}$
Since, the denominator has only 5 as its factor,
thus, ${}^{6}/{}_{15}$ has a terminating decimal expansion.
(ix) ${}^{35}/{}_{50}$
${}^{35}/{}_{50}={}^{7}/{}_{10}$
Factorising the denominator, we get,
$10=2\times 5$
The denominator is in the form of ${{2}^{m}}\times {{5}^{n}}$ where m and n are non negative integers
Hence, ${}^{35}/{}_{50}$ has a terminating decimal expansion.
(x) ${}^{77}/{}_{210}$
${}^{77}/{}_{210}$=${}^{7\times 11}/{}_{7\times 30}$ $={}^{11}/{}_{30}$
Factorising the denominator, we get,
$30=2\times 3\times 5$
The denominator is not in the form of ${{2}^{m}}\times {{5}^{n}}$ .where m and n are non negative integers.
Hence, ${}^{77}/{}_{210}$ has a non-terminating repeating decimal expansion.