Answer:
[tex]=\frac{27}{125}[/tex]
Step-by-step explanation:
[tex]0.216[/tex]
[tex]\mathrm{Multiply\:and\:divide\:by\:10\:for\:every\:number\:after\:the\:decimal\:point.}[/tex]
[tex]\mathrm{There\:are\:}3\mathrm{\:digits\:to\:the\:right\:of\:the\:decimal\:point,\:therefore\:multiply\:and\:divide\:by\:}1000[/tex]
[tex]=\frac{0.216\cdot \:1000}{1000}[/tex]
[tex]=\frac{216}{1000}[/tex]
[tex]216,\:1000[/tex]
[tex]\mathrm{The\:GCD\:of\:}a,\:b\mathrm{\:is\:the\:largest\:positive\:number\:that\:divides\:both\:}a\mathrm{\:and\:}b\mathrm{\:without\:a\:remainder}[/tex]
[tex]2,\:3\mathrm{\:are\:all\:prime\:numbers,\:therefore\:no\:further\:factorization\:is\:possible}[/tex]
[tex]=2\cdot \:2\cdot \:2\cdot \:3\cdot \:3\cdot \:3[/tex]
[tex]2,\:5\mathrm{\:are\:all\:prime\:numbers,\:therefore\:no\:further\:factorization\:is\:possible}[/tex]
[tex]=2\cdot \:2\cdot \:2\cdot \:5\cdot \:5\cdot \:5[/tex]
[tex]\mathrm{The\:prime\:factors\:common\:to\:}216,\:1000\mathrm{\:are\:}[/tex]
[tex]=2\cdot \:2\cdot \:2[/tex]
[tex]\mathrm{Multiply\:the\:numbers:}\:2\cdot \:2\cdot \:2=8[/tex]
[tex]=8[/tex]
[tex]=\frac{8\cdot \:27}{8\cdot \:125}[/tex]
[tex]\mathrm{Cancel\:the\:common\:factor:}\:8[/tex]
[tex]=\frac{27}{125}[/tex]
Hence, the correct answer is [tex]=\frac{27}{125}[/tex]
