Question

without actual division show that each of the following rational number is a terminating decimal Express each in decimal form first 20352 power 3 into 5 power 2 second 24 by 125 third 171/800 fourth 15/1600 fifth 17/320 sixth 19/3125​

Answer 1

Step-by-step explanation:

Without actual division show that each of the following rational number is a terminating decimal Express each in decimal form

$1) \frac{ 20352}{ {3 \times 5}^{2} } \\ \\ 2) \frac{24}{125} \\ \\ 3) \frac{171}{800} \\ \\ 4) \frac{15}{1600} \\ \\ 5) \frac{19}{3125}$

To solve these questions:

1) Calculate the prime factors of denominator

2) If those factors are in the form

${2}^{n} \times {5}^{m} \: \: \: \: m \:, n \: \geqslant 0$

then the numbers have terminating decimal expansion.

$1) \frac{ (20352) }{3 \times ( {5)}^{2} } \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: not \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ but \: the \: given \: number \: is \: not \: simple \: rational \: number \\ \: so \: convert \: it \: into \: simple \: form \: first \\ \\ \frac{6784}{ {5}^{2} } \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{6784 \times 4}{ {(2 \times 5)}^{2} } \\ \\ = \frac{27136}{( {10)}^{2} } \\ \\ = \frac{27136}{100} = 271.36$

$2) \: \frac{24}{125} = \frac{24}{ {(5)}^{3} } \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{24 \times {(2}^{3} )}{( {5}^{3})( {2}^{3} ) } \\ \\ = \frac{24 \times8 }{( {10)}^{3} } \\ \\ = \frac{192}{1000} = 0.192$

$3) \frac{171}{800} \\ \\ = \frac{171}{( {2}^{5})( {5}^{2} )} \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{171 \times {(5}^{3} )}{( {2}^{5})( {5}^{5} ) } \\ \\ = \frac{171 \times125 }{( {10)}^{5} } \\ \\ = \frac{21375}{100000} = 0.21375$

$4) \frac{15}{1600} \\ \\ = \frac{15}{ {2}^{6} \times {5}^{2} } \\ \\ = \frac{3}{ {2}^{6} \times 5} \\ \\ = \frac{3 \times {5}^{5} }{ {2}^{6} \times {5}^{6} } \\ \\ = \frac{3 \times3125 }{( {10)}^{6} } \\ \\ = \frac{9375}{1000000} \\ \\ = 0.009375 \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion.$

$5) \frac{17}{320} \\ \\ \frac{17}{ {2}^{6} \times 5 } it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{17 \times {(5}^{5} )}{( {2}^{6})( {5}^{6} ) } \\ \\ = \frac{17\times \: 3125}{( {10)}^{6} } \\ \\ = \frac{53125}{1000000} = 0.053125$

$6) \frac{19}{3125} \\ \\ = \frac{19}{ {5}^{5} } \\ \\ it \: is \: \: shown \: that \: factors \: of \: denominator \: are \: \: in \: \\ form \: {2}^{n} \times {5}^{m} \\ \\ this \: rational \: number \: have \: terminating \: decimal \: expansion. \\ \\ \frac{19 \times {(2}^{5} )}{( {5}^{5})( {2}^{5} ) } \\ \\ = \frac{19 \times \: 32 }{( {10)}^{5} } \\ \\ = \frac{608}{100000} = 0.00608$

Hope it helps you.