Without actual division show that each of the following rational numbers is a terminating decimal express it in decimal form
A.23/(2³*5²)
B.24/125
C.171/800
D.15/1600
E.17/320
F.19/3125
Answers
Step-by-step explanation:
Without actual division show that each of the following rational numbers is a terminating decimal Express each in decimal form
\begin{gathered}1) \frac{ 20352}{ {3 \times 5}^{2} } \\ \\ 2) \frac{24}{125} \\ \\ 3) \frac{171}{800} \\ \\ 4) \frac{15}{1600} \\ \\ 5) \frac{19}{3125} \\ \\\end{gathered}
1)
3×5
2
20352
2)
125
24
3)
800
171
4)
1600
15
5)
3125
19
To solve these questions:
1) Calculate the prime factors of the denominator
2) If those factors are in the form
{2}^{n} \times {5}^{m} \: \: \: \: m \:, n \: \geqslant 02
n
×5
m
m,n⩾0
then the numbers have terminating decimal expansion.
\begin{gathered}1) \frac{ (20352) }{3 \times ( {5)}^{2} } \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: not \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ but \: the \: given \: number \: is \: not \: simple \: rational \: number \\ \: so \: convert \: it \: into \: simple \: form \: first \\ \\ \frac{6784}{ {5}^{2} } \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{6784 \times 4}{ {(2 \times 5)}^{2} } \\ \\ = \frac{27136}{( {10)}^{2} } \\ \\ = \frac{27136}{100} = 271.36 \\ \\\end{gathered}
1)
3×(5)
2
(20352)
itisshownthatfactorsofdenominatorarenotin
form2
n
×5
m
butthegivennumberisnotsimplerationalnumber
soconvertitintosimpleformfirst
5
2
6784
thisrationalnumberhaveterminatingdecimalexpansion.
(2×5)
2
6784×4
=
(10)
2
27136
=
100
27136
=271.36
\begin{gathered}2) \: \frac{24}{125} = \frac{24}{ {(5)}^{3} } \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{24 \times {(2}^{3} )}{( {5}^{3})( {2}^{3} ) } \\ \\ = \frac{24 \times8 }{( {10)}^{3} } \\ \\ = \frac{192}{1000} = 0.192 \\ \\\end{gathered}
2)
125
24
=
(5)
3
24
itisshownthatfactorsofdenominatorarein
form2
n
×5
m
thisrationalnumberhaveterminatingdecimalexpansion.
(5
3
)(2
3
)
24×(2
3
)
=
(10)
3
24×8
=
1000
192
=0.192
\begin{gathered}3) \frac{171}{800} \\ \\ = \frac{171}{( {2}^{5})( {5}^{2} )} \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{171 \times {(5}^{3} )}{( {2}^{5})( {5}^{5} ) } \\ \\ = \frac{171 \times125 }{( {10)}^{5} } \\ \\ = \frac{21375}{100000} = 0.21375 \\ \\\end{gathered}
3)
800
171
=
(2
5
)(5
2
)
171
itisshownthatfactorsofdenominatorarein
form2
n
×5
m
thisrationalnumberhaveterminatingdecimalexpansion.
(2
5
)(5
5
)
171×(5
3
)
=
(10)
5
171×125
=
100000
21375
=0.21375
\begin{gathered}4) \frac{15}{1600} \\ \\ = \frac{15}{ {2}^{6} \times {5}^{2} } \\ \\ = \frac{3}{ {2}^{6} \times 5} \\ \\ = \frac{3 \times {5}^{5} }{ {2}^{6} \times {5}^{6} } \\ \\ = \frac{3 \times3125 }{( {10)}^{6} } \\ \\ = \frac{9375}{1000000} \\ \\ = 0.009375 \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\\end{gathered}
4)
1600
15
=
2
6
×5
2
15
=
2
6
×5
3
=
2
6
×5
6
3×5
5
=
(10)
6
3×3125
=
1000000
9375
=0.009375
itisshownthatfactorsofdenominatorarein
form2
n
×5
m
thisrationalnumberhaveterminatingdecimalexpansion.
\begin{gathered}5) \frac{17}{320} \\ \\ \frac{17}{ {2}^{6} \times 5 } it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{17 \times {(5}^{5} )}{( {2}^{6})( {5}^{6} ) } \\ \\ = \frac{17\times \: 3125}{( {10)}^{6} } \\ \\ = \frac{53125}{1000000} = 0.053125 \\ \\\end{gathered}
5)
320
17
2
6
×5
17
itisshownthatfactorsofdenominatorarein
form2
n
×5
m
thisrationalnumberhaveterminatingdecimalexpansion.
(2
6
)(5
6
)
17×(5
5
)
=
(10)
6
17×3125
=
1000000
53125
=0.053125
\begin{gathered}6) \frac{19}{3125} \\ \\ = \frac{19}{ {5}^{5} } \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{19 \times {(2}^{5} )}{( {5}^{5})( {2}^{5} ) } \\ \\ = \frac{19 \times \: 32 }{( {10)}^{5} } \\ \\ = \frac{608}{100000} = 0.00608 \\ \\\end{gathered}
6)
3125
19
=
5
5
19
itisshownthatfactorsofdenominatorarein
form2
n
×5
m
this rational number has a terminating decimal expansion.
(5
5
)(2
5
)
19×(2
5
)
=
(10)
5
19×32
=
100000
608
=0.00608
Hope it helps you.