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Suku yang bebas p pada penjabaran (2p+p23​)12 adalah ...

Pertanyaan

Suku yang bebas begin mathsize 14px style p end style pada penjabaran begin mathsize 14px style open parentheses 2 p plus 3 over p squared close parentheses to the power of 12 end style adalah ...

  1. begin mathsize 14px style 10.156.320 end style 

  2. begin mathsize 14px style 10.240.320 end style 

  3. begin mathsize 14px style 10.264.320 end style 

  4. begin mathsize 14px style 10.371.240 end style 

  5. begin mathsize 14px style 10.394.120 end style 

Pembahasan:

Rumus Binomial Newton adalah sebagai berikut.

open parentheses a plus b close parentheses to the power of n equals sum from i equals 0 to n of open parentheses table row n row i end table close parentheses a to the power of open parentheses n minus i close parentheses end exponent b to the power of i

Dengan

a equals 2 p b equals 3 over p squared

Maka,

  • i equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 12 factorial over denominator 0 factorial open parentheses 12 minus 0 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 12 open parentheses 3 over p squared close parentheses to the power of 0 end cell equals cell fraction numerator 12 factorial over denominator 0 factorial open parentheses 12 minus 0 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 12 times 1 end cell row blank equals cell fraction numerator 12 factorial over denominator 0 factorial open parentheses 12 minus 0 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 12 end cell row blank equals cell fraction numerator 12 factorial over denominator 12 factorial end fraction times open parentheses 2 p close parentheses to the power of 12 end cell row blank equals cell 1 times 2 to the power of 12 p to the power of 12 end cell row blank equals cell 4.096 p to the power of 12 end cell row blank blank blank end table

  • i equals 1

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 12 factorial over denominator 1 factorial open parentheses 12 minus 1 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 11 open parentheses 3 over p squared close parentheses to the power of 1 end cell equals cell fraction numerator 12 factorial over denominator 1 factorial times thin space 11 factorial end fraction open parentheses 2 p close parentheses to the power of 11 open parentheses 3 over p squared close parentheses end cell row blank equals cell fraction numerator 12 times up diagonal strike 11 factorial end strike over denominator up diagonal strike 11 factorial end strike end fraction times 2 to the power of 11 p to the power of 11 times open parentheses 3 over p squared close parentheses end cell row blank equals cell fraction numerator 12 times thin space 3 times thin space 2 to the power of 11 p to the power of 11 over denominator p squared end fraction end cell row blank equals cell 2 to the power of 11 times thin space 36 p to the power of 11 minus 2 end exponent end cell row blank equals cell 2 to the power of 11 times 36 to the power of 9 end cell row blank equals cell 2.048 times thin space 36 p to the power of 9 end cell row blank equals cell 73.728 p to the power of 9 end cell end table

  • i equals 2

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 12 factorial over denominator 2 factorial open parentheses 12 minus 2 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 10 open parentheses 3 over p squared close parentheses squared end cell equals cell fraction numerator 12 factorial over denominator 2 factorial times thin space 10 factorial end fraction end cell row blank equals cell fraction numerator 12 times 11 times up diagonal strike 10 factorial end strike over denominator 2 factorial times thin space up diagonal strike 10 factorial end strike end fraction times 2 to the power of 10 times p to the power of 10 times open parentheses 3 over p squared close parentheses squared end cell row blank equals cell fraction numerator 12 times thin space 11 over denominator 2 factorial end fraction times 2 to the power of 10 times p to the power of 10 times open parentheses 3 squared over p to the power of 4 to the power of blank end exponent close parentheses end cell row blank equals cell fraction numerator 12 times thin space 11 over denominator 2 factorial end fraction times fraction numerator 132 times thin space 3 squared times thin space 2 to the power of 10 p to the power of 10 over denominator p to the power of 4 end fraction end cell row blank equals cell fraction numerator 2 to the power of 10 times thin space 3 squared times thin space 132 p to the power of 10 minus 4 end exponent over denominator 2 factorial end fraction end cell row blank equals cell fraction numerator 1.024 times 9 times 132 p to the power of 6 over denominator 2 times 1 end fraction end cell row blank equals cell fraction numerator 1.216.512 p to the power of 6 over denominator 2 end fraction end cell row blank equals cell 606.256 p to the power of 6 end cell end table

  • i equals 3

fraction numerator 12 factorial over denominator 3 factorial open parentheses 12 minus 3 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 9 open parentheses 3 over p squared close parentheses cubed equals fraction numerator 12 factorial over denominator 3 factorial times thin space 9 factorial end fraction times 2 to the power of 9 p to the power of 9 times 3 cubed over p to the power of 6 p to the power of 9 equals fraction numerator 12 times thin space 11 times thin space 10 times up diagonal strike 9 factorial end strike over denominator 3 factorial times up diagonal strike 9 factorial end strike end fraction times 2 to the power of 9 p to the power of 9 times 3 cubed over p to the power of 6 p to the power of 9 equals fraction numerator 12 times thin space 11 times thin space 10 over denominator 3 factorial end fraction times times 2 to the power of 9 p to the power of 9 times 3 cubed over p to the power of 6 p to the power of 9 equals fraction numerator 1.320 times thin space 3 cubed times thin space 2 to the power of 9 p to the power of 9 over denominator 3 factorial p to the power of 6 end fraction equals fraction numerator 2 to the power of 9 times thin space 3 cubed times thin space 1.320 p to the power of 9 minus 6 end exponent over denominator 3 factorial end fraction equals fraction numerator 2 to the power of 9 times thin space 3 cubed times thin space 1.320 p cubed over denominator 3 factorial end fraction equals fraction numerator 512 times 27 times 1.320 p squared over denominator 3 times 2 times 1 end fraction equals fraction numerator 18.247.680 p cubed over denominator 6 end fraction equals 3.041.280 p cubed

 

  • i equals 4

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 12 factorial over denominator 4 factorial open parentheses 12 minus 4 close parentheses factorial end fraction open parentheses 2 p close parentheses to the power of 8 open parentheses 3 over p squared close parentheses to the power of 4 end cell equals cell fraction numerator 12 factorial over denominator 4 factorial open parentheses 12 minus 4 close parentheses factorial end fraction times 2 to the power of 8 p to the power of 8 times 3 to the power of 4 over open parentheses p squared close parentheses to the power of 4 end cell row blank equals cell fraction numerator 12 factorial over denominator 4 factorial times thin space 8 factorial end fraction times 2 to the power of 8 p to the power of 8 times 3 to the power of 4 over p to the power of 8 end cell row blank equals cell fraction numerator 12 times thin space 11 times thin space 10 times thin space 9 times up diagonal strike 8 factorial end strike over denominator 4 factorial times up diagonal strike 8 factorial end strike end fraction times 2 to the power of 8 p to the power of 8 times 3 to the power of 4 over p to the power of 8 end cell row blank equals cell fraction numerator 12 times thin space 11 times thin space 10 times thin space 9 over denominator 4 factorial end fraction times 2 to the power of 8 p to the power of 8 times 3 to the power of 4 over p to the power of 8 end cell row blank equals cell fraction numerator 11.880 over denominator 4 factorial end fraction times 2 to the power of 8 times 3 to the power of 4 over p to the power of 8 p to the power of 8 end cell row blank equals cell fraction numerator 11.880 over denominator 4 times 3 times 2 times 1 end fraction times 2 to the power of 8 times fraction numerator 3 to the power of 4 over denominator up diagonal strike p to the power of 8 end strike end fraction up diagonal strike p to the power of 8 end strike end cell row blank equals cell fraction numerator 11.880 over denominator 24 end fraction times 256 times 81 end cell row blank equals cell 495 times 256 times 81 end cell row blank equals cell 10.264.320 end cell end table

 

Jadi, suku yang bebas begin mathsize 14px style p end style pada penjabaran begin mathsize 14px style open parentheses 2 p plus 3 over p squared close parentheses to the power of 12 end style adalah begin mathsize 14px style 10.264.320 end style .

 

Oleh karena itu, jawaban yang tepat adalah C.

Jawaban terverifikasi

Dijawab oleh:

G. Albiah

Mahasiswa/Alumni Universitas Galuh Ciamis

Terakhir diupdate 07 Oktober 2021

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