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Hasil dari  adalah

Pertanyaan

Hasil dari 2 square root of 150 plus 5 square root of 54 minus 7 square root of 96 adalah horizontal ellipsis 

Pembahasan Video:

Pembahasan Soal:

Misalkan: M equals 2 square root of 150 plus 5 square root of 54 minus 7 square root of 96. Dengan menggunakan sifat bentuk akar dan aturan operasi hitung campuran, maka hasil dari M adalah

table attributes columnalign right center left columnspacing 0px end attributes row M equals cell 2 square root of 150 plus 5 square root of 54 minus 7 square root of 96 end cell row blank equals cell 2 square root of 25 cross times 6 end root plus 5 square root of 9 cross times 6 end root minus 7 square root of 16 cross times 6 end root end cell row blank equals cell 2 times 5 square root of 6 plus 5 times 3 square root of 6 minus 7 times 4 square root of 6 end cell row blank equals cell 10 square root of 6 plus 15 square root of 6 minus 28 square root of 6 end cell row blank equals cell open parentheses 10 plus 15 minus 28 close parentheses square root of 6 end cell row blank equals cell open parentheses 25 minus 28 close parentheses square root of 6 end cell row M equals cell negative 3 square root of 6 end cell end table

Dengan demikian, hasil dari 2 square root of 150 plus 5 square root of 54 minus 7 square root of 96 adalah negative 3 square root of 6. 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 13 September 2021

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Tentukan hasil operasi bilangan berikut! d.  e.  f.  g.

Pembahasan Soal:

d. Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga 5 square root of 12 plus 4 square root of 75 diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 square root of 12 plus 4 square root of 75 end cell equals cell 5 square root of 4 cross times 3 end root minus 4 square root of 25 cross times 3 end root end cell row blank equals cell 5 square root of 2 squared cross times 3 end root minus 4 square root of 5 squared cross times 3 end root end cell row blank equals cell 5 cross times 2 square root of 3 minus 4 cross times 5 square root of 3 end cell row blank equals cell 10 square root of 3 minus 20 square root of 3 end cell end table

Berdasarkan aturan pengurangan bentuk akar dengan bilangan pokok sama yaitu a n-th root of b minus c n-th root of b equals open parentheses a minus c close parentheses n-th root of b maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 square root of 12 plus 4 square root of 75 end cell equals cell 5 square root of 4 cross times 3 end root minus 4 square root of 25 cross times 3 end root end cell row blank equals cell 5 square root of 2 squared cross times 3 end root minus 4 square root of 5 squared cross times 3 end root end cell row blank equals cell 5 cross times 2 square root of 3 minus 4 cross times 5 square root of 3 end cell row blank equals cell 10 square root of 3 minus 20 square root of 3 end cell end table

Jadi hasil dari 5 square root of 12 plus 4 square root of 75 adalah negative 10 square root of 3.

e. Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga 3 cube root of 40 plus 4 cube root of 320 minus 2 cube root of 125 diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 cube root of 40 plus 4 cube root of 320 minus 2 cube root of 125 end cell equals cell 3 cube root of 8 cross times 5 end root plus 4 cube root of 64 cross times 5 end root minus 2 cube root of 5 cubed end root end cell row blank equals cell 3 cube root of 2 cubed cross times 5 end root plus 4 cube root of 4 cubed cross times 5 end root minus 2 cube root of 5 cubed end root end cell row blank equals cell 3 cross times 2 cube root of 5 plus 4 cross times 4 cube root of 5 minus 2 cross times 5 end cell row blank equals cell 6 cube root of 5 plus 16 cube root of 5 minus 10 end cell end table

Berdasarkan aturan penjumlahan bentuk akar dengan bilangan pokok sama yaitu a n-th root of b plus c n-th root of b equals open parentheses a plus c close parentheses n-th root of b maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 cube root of 40 plus 4 cube root of 320 minus 2 cube root of 125 end cell equals cell 3 cube root of 8 cross times 5 end root plus 4 cube root of 64 cross times 5 end root minus 2 cube root of 5 cubed end root end cell row blank equals cell 3 cube root of 2 cubed cross times 5 end root plus 4 cube root of 4 cubed cross times 5 end root minus 2 cube root of 5 cubed end root end cell row blank equals cell 3 cross times 2 cube root of 5 plus 4 cross times 4 cube root of 5 minus 2 cross times 5 end cell row blank equals cell 6 cube root of 5 plus 16 cube root of 5 minus 10 end cell row blank equals cell open parentheses 6 plus 16 close parentheses cube root of 5 minus 10 end cell row blank equals cell 22 cube root of 5 minus 10 end cell end table

Jadi hasil dari 3 cube root of 40 plus 4 cube root of 320 minus 2 cube root of 125 adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 22 cube root of 5 minus 10 end cell end table.

f. Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga 6 square root of 20 plus 2 square root of 75 minus 4 square root of 5 plus 3 square root of 48 diperoleh sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 6 square root of 20 plus 2 square root of 75 minus 4 square root of 5 plus 3 square root of 48 end cell equals cell 6 square root of 4 cross times 5 end root plus 2 square root of 25 cross times 3 end root minus 4 square root of 5 plus 3 square root of 16 cross times 3 end root end cell row blank equals cell 6 square root of 2 squared cross times 5 end root plus 2 square root of 5 squared cross times 3 end root minus 4 square root of 5 plus 3 square root of 4 squared cross times 3 end root end cell row blank equals cell 6 cross times 2 square root of 5 plus 2 cross times 5 square root of 3 minus 4 square root of 5 plus 3 cross times 4 square root of 3 end cell row blank equals cell 12 square root of 5 plus 10 square root of 3 minus 4 square root of 5 plus 12 square root of 3 end cell end table end style

Berdasarkan aturan penjumlahan dan pengurangan bentuk akar dengan bilangan pokok sama yaitu a n-th root of b plus-or-minus c n-th root of b equals open parentheses a plus-or-minus c close parentheses n-th root of b maka

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 6 square root of 20 plus 2 square root of 75 minus 4 square root of 5 plus 3 square root of 48 end cell equals cell 6 square root of 4 cross times 5 end root plus 2 square root of 25 cross times 3 end root minus 4 square root of 5 plus 3 square root of 16 cross times 3 end root end cell row blank equals cell 6 square root of 2 squared cross times 5 end root plus 2 square root of 5 squared cross times 3 end root minus 4 square root of 5 plus 3 square root of 4 squared cross times 3 end root end cell row blank equals cell 6 cross times 2 square root of 5 plus 2 cross times 5 square root of 3 minus 4 square root of 5 plus 3 cross times 4 square root of 3 end cell row blank equals cell 12 square root of 5 plus 10 square root of 3 minus 4 square root of 5 plus 12 square root of 3 end cell row blank equals cell 12 square root of 5 minus 4 square root of 5 plus 10 square root of 3 plus 12 square root of 3 end cell row blank equals cell open parentheses 12 minus 4 close parentheses square root of 5 plus open parentheses 10 plus 12 close parentheses square root of 3 end cell row blank equals cell 8 square root of 5 plus 22 square root of 3 end cell end table end style

Jadi hasil dari 6 square root of 20 plus 2 square root of 75 minus 4 square root of 5 plus 3 square root of 48 adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 8 square root of 5 plus 22 square root of 3 end cell end table.

g. Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga 4 square root of 24 minus 5 square root of 96 plus 2 cube root of 16 plus 7 cube root of 54 diperoleh sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 4 square root of 24 minus 5 square root of 96 plus 2 cube root of 16 plus 7 cube root of 54 end cell equals cell 4 square root of 4 cross times 6 end root minus 5 square root of 16 cross times 6 end root plus 2 cube root of 8 cross times 2 end root plus 7 cube root of 27 cross times 2 end root end cell row blank equals cell 4 square root of 2 squared cross times 6 end root minus 5 square root of 4 squared cross times 6 end root plus 2 cube root of 2 cubed cross times 2 end root plus 7 cube root of 3 cubed cross times 2 end root end cell row blank equals cell 4 cross times 2 square root of 6 minus 5 cross times 4 square root of 6 plus 2 cross times 2 cube root of 2 plus 7 cross times 3 cube root of 2 end cell row blank equals cell 8 square root of 6 minus 20 square root of 6 plus 4 cube root of 2 plus 21 cube root of 2 end cell end table end style

Berdasarkan aturan penjumlahan dan pengurangan bentuk akar dengan bilangan pokok sama yaitu a n-th root of b plus-or-minus c n-th root of b equals open parentheses a plus-or-minus c close parentheses n-th root of b maka

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 4 square root of 24 minus 5 square root of 96 plus 2 cube root of 16 plus 7 cube root of 54 end cell equals cell 4 square root of 4 cross times 6 end root minus 5 square root of 16 cross times 6 end root plus 2 cube root of 8 cross times 2 end root plus 7 cube root of 27 cross times 2 end root end cell row blank equals cell 4 square root of 2 squared cross times 6 end root minus 5 square root of 4 squared cross times 6 end root plus 2 cube root of 2 cubed cross times 2 end root plus 7 cube root of 3 cubed cross times 2 end root end cell row blank equals cell 4 cross times 2 square root of 6 minus 5 cross times 4 square root of 6 plus 2 cross times 2 cube root of 2 plus 7 cross times 3 cube root of 2 end cell row blank equals cell 8 square root of 6 minus 20 square root of 6 plus 4 cube root of 2 plus 21 cube root of 2 end cell row blank equals cell open parentheses 8 minus 20 close parentheses square root of 6 plus open parentheses 4 plus 21 close parentheses cube root of 2 end cell row blank equals cell negative 12 square root of 6 plus 25 cube root of 2 end cell end table end style

Jadi hasil dari 4 square root of 24 minus 5 square root of 96 plus 2 cube root of 16 plus 7 cube root of 54 adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative 12 square root of 6 plus 25 cube root of 2 end cell end table.

 

0

Roboguru

Hasil dari  adalah

Pembahasan Soal:

Ingat jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p text  atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan.

Diketahui square root of 80 space text dan  end text square root of 45 merupakan bilangan akar pangkat dua sehingga square root of 80 minus 3 square root of 45 plus 3 square root of 5 diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 80 minus 3 square root of 45 plus 3 square root of 5 end cell equals cell square root of 16 cross times 5 end root minus 3 square root of 9 cross times 5 end root plus 3 square root of 5 end cell row blank equals cell square root of 4 squared cross times 5 end root minus 3 square root of 3 squared cross times 5 end root plus 3 square root of 5 end cell row blank equals cell 4 square root of 5 minus 3 cross times 3 square root of 5 plus 3 square root of 5 end cell row blank equals cell 4 square root of 5 minus 9 square root of 5 plus 3 square root of 5 end cell row blank equals cell open parentheses 4 minus 9 plus 3 close parentheses square root of 5 end cell row blank equals cell negative 2 square root of 5 end cell end table

Jadi hasil dari square root of 80 minus 3 square root of 45 plus 3 square root of 5 adalah negative 2 square root of 5.

Oleh karena itu, jawaban yang benar adalah B.

4

Roboguru

Bentuk sederhana dari  adalah ...

Pembahasan Soal:

Sederhanakan bentuk begin mathsize 14px style fraction numerator 7 square root of 27 plus 3 square root of 147 over denominator 9 square root of 12 minus 3 square root of 243 end fraction end style terlebih dahulu supaya mempunyai bilangan pokok yang sama, kemudian gunakan sifat-sifat bentuk akar:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 7 square root of 27 plus 3 square root of 147 over denominator 9 square root of 12 minus 3 square root of 243 end fraction end cell equals cell fraction numerator 7 square root of 9 cross times 3 end root plus 3 square root of 49 cross times 3 end root over denominator 9 square root of 4 cross times 3 end root minus 3 square root of 81 cross times 3 end root end fraction end cell row blank equals cell fraction numerator open parentheses 7 square root of 9 cross times square root of 3 close parentheses plus open parentheses 3 square root of 49 cross times square root of 3 close parentheses over denominator open parentheses 9 square root of 4 cross times square root of 3 close parentheses minus open parentheses 3 square root of 81 cross times square root of 3 close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses 7 cross times 3 cross times square root of 3 close parentheses plus open parentheses 3 cross times 7 cross times square root of 3 close parentheses over denominator open parentheses 9 cross times 2 cross times square root of 3 close parentheses minus open parentheses 3 cross times 9 cross times square root of 3 close parentheses end fraction end cell row blank equals cell fraction numerator 21 square root of 3 plus 21 square root of 3 over denominator 18 square root of 3 minus 27 square root of 3 end fraction end cell row blank equals cell fraction numerator open parentheses 21 plus 21 close parentheses square root of 3 over denominator open parentheses 18 minus 27 close parentheses square root of 3 end fraction end cell row blank equals cell fraction numerator 42 square root of 3 over denominator negative 9 square root of 3 end fraction end cell row blank equals cell fraction numerator 14 over denominator negative 3 end fraction end cell end table 

Oleh karena itu jawaban yang benar adalah A.

0

Roboguru

Rasionalkan penyebut pecahan-pecahan berikut! e.  f.  g.  h.

Pembahasan Soal:

e. Diketahui fraction numerator 5 over denominator square root of 12 minus 3 end fraction maka cara untuk merasionalkannya adalah dengan mengalikan akar sekawan dari square root of 12 minus 3 yaitu square root of 12 plus 3 sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction end cell equals cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction cross times fraction numerator square root of 12 plus 3 over denominator square root of 12 plus 3 end fraction end cell row blank equals cell fraction numerator 5 open parentheses square root of 12 plus 3 close parentheses over denominator open parentheses square root of 12 minus 3 close parentheses open parentheses square root of 12 plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 5 cross times square root of 12 plus 5 cross times 3 over denominator square root of 12 cross times square root of 12 plus 3 cross times square root of 12 minus 3 cross times square root of 12 minus 3 cross times 3 end fraction end cell end table

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu n-th root of a cross times n-th root of b equals n-th root of a cross times b end root maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction end cell equals cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction cross times fraction numerator square root of 12 plus 3 over denominator square root of 12 plus 3 end fraction end cell row blank equals cell fraction numerator 5 open parentheses square root of 12 plus 3 close parentheses over denominator open parentheses square root of 12 minus 3 close parentheses open parentheses square root of 12 plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 5 cross times square root of 12 plus 5 cross times 3 over denominator square root of 12 cross times square root of 12 plus 3 cross times square root of 12 minus 3 cross times square root of 12 minus 3 cross times 3 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator square root of 12 cross times 12 end root plus up diagonal strike 3 square root of 12 minus 3 square root of 12 end strike minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator square root of 12 squared end root minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator 12 minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator 3 end fraction end cell end table

Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction end cell equals cell fraction numerator 5 over denominator square root of 12 minus 3 end fraction cross times fraction numerator square root of 12 plus 3 over denominator square root of 12 plus 3 end fraction end cell row blank equals cell fraction numerator 5 open parentheses square root of 12 plus 3 close parentheses over denominator open parentheses square root of 12 minus 3 close parentheses open parentheses square root of 12 plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 5 cross times square root of 12 plus 5 cross times 3 over denominator square root of 12 cross times square root of 12 plus 3 cross times square root of 12 minus 3 cross times square root of 12 minus 3 cross times 3 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator square root of 12 cross times 12 end root plus up diagonal strike 3 square root of 12 minus 3 square root of 12 end strike minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator square root of 12 squared end root minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator 12 minus 9 end fraction end cell row blank equals cell fraction numerator 5 square root of 12 plus 15 over denominator 3 end fraction end cell row blank equals cell fraction numerator 5 square root of 4 cross times 3 end root plus 15 over denominator 3 end fraction end cell row blank equals cell fraction numerator 5 square root of 2 squared cross times 3 end root plus 15 over denominator 3 end fraction end cell row blank equals cell fraction numerator 5 cross times 2 square root of 3 plus 15 over denominator 3 end fraction end cell row blank equals cell fraction numerator 10 square root of 3 plus 15 over denominator 3 end fraction end cell end table

Jadi bentuk rasional dari fraction numerator 5 over denominator square root of 12 minus 3 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 10 square root of 3 plus 15 over denominator 3 end fraction end cell end table.

f. Diketahui fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction maka cara merasionalkannya adalah dengan mengalikan penyebut akar dengan akar sekawannya yaitu square root of 7 minus square root of 6 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction cross times fraction numerator square root of 7 minus square root of 6 over denominator square root of 7 minus square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 open parentheses square root of 7 minus square root of 6 close parentheses over denominator open parentheses square root of 7 plus square root of 6 close parentheses open parentheses square root of 7 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times square root of 7 minus 6 square root of 3 cross times square root of 6 over denominator square root of 7 cross times square root of 7 minus square root of 7 cross times square root of 6 plus square root of 6 cross times square root of 7 minus square root of 6 cross times square root of 6 end fraction end cell end table

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu n-th root of a cross times n-th root of b equals n-th root of a cross times b end root maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction cross times fraction numerator square root of 7 minus square root of 6 over denominator square root of 7 minus square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 open parentheses square root of 7 minus square root of 6 close parentheses over denominator open parentheses square root of 7 plus square root of 6 close parentheses open parentheses square root of 7 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times square root of 7 minus 6 square root of 3 cross times square root of 6 over denominator square root of 7 cross times square root of 7 minus square root of 7 cross times square root of 6 plus square root of 6 cross times square root of 7 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times 7 end root minus 6 square root of 3 cross times 6 end root over denominator square root of 7 cross times 7 end root minus square root of 7 cross times 6 end root plus square root of 6 cross times 7 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 18 over denominator square root of 49 up diagonal strike negative square root of 42 plus square root of 42 end strike minus square root of 36 end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 18 over denominator square root of 49 minus square root of 36 end fraction end cell end table

Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction cross times fraction numerator square root of 7 minus square root of 6 over denominator square root of 7 minus square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 open parentheses square root of 7 minus square root of 6 close parentheses over denominator open parentheses square root of 7 plus square root of 6 close parentheses open parentheses square root of 7 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times square root of 7 minus 6 square root of 3 cross times square root of 6 over denominator square root of 7 cross times square root of 7 minus square root of 7 cross times square root of 6 plus square root of 6 cross times square root of 7 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 3 cross times 7 end root minus 6 square root of 3 cross times 6 end root over denominator square root of 7 cross times 7 end root minus square root of 7 cross times 6 end root plus square root of 6 cross times 7 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 18 over denominator square root of 49 up diagonal strike negative square root of 42 plus square root of 42 end strike minus square root of 36 end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 18 over denominator square root of 49 minus square root of 36 end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 9 cross times 2 end root over denominator 7 minus 6 end fraction end cell row blank equals cell fraction numerator 6 square root of 21 minus 6 square root of 3 squared cross times 2 end root over denominator 1 end fraction end cell row blank equals cell 6 square root of 21 minus 6 cross times 3 square root of 2 end cell row blank equals cell 6 square root of 21 minus 18 square root of 2 end cell end table

Jadi bentuk rasional dari fraction numerator 6 square root of 3 over denominator square root of 7 plus square root of 6 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 6 square root of 21 minus 18 square root of 2 end cell end table.

g. Diketahui fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction maka cara merasionalkannya adalah dengan mengalikan penyebut dengan akar sekawannya yaitu square root of 15 plus square root of 8 sehingga 

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction end cell equals cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction cross times fraction numerator square root of 15 plus square root of 8 over denominator square root of 15 plus square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 plus square root of 5 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses over denominator open parentheses square root of 15 minus square root of 8 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses end fraction end cell row blank equals cell fraction numerator 4 cross times square root of 15 plus 4 cross times square root of 8 plus square root of 5 cross times square root of 15 plus square root of 5 cross times square root of 8 over denominator square root of 15 cross times square root of 15 plus square root of 15 cross times square root of 8 minus square root of 8 cross times square root of 15 minus square root of 8 cross times square root of 8 end fraction end cell end table

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu n-th root of a cross times n-th root of b equals n-th root of a cross times b end root maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction end cell equals cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction cross times fraction numerator square root of 15 plus square root of 8 over denominator square root of 15 plus square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 plus square root of 5 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses over denominator open parentheses square root of 15 minus square root of 8 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses end fraction end cell row blank equals cell fraction numerator 4 cross times square root of 15 plus 4 cross times square root of 8 plus square root of 5 cross times square root of 15 plus square root of 5 cross times square root of 8 over denominator square root of 15 cross times square root of 15 plus square root of 15 cross times square root of 8 minus square root of 8 cross times square root of 15 minus square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 5 cross times 15 end root plus square root of 5 cross times 8 end root over denominator square root of 15 cross times 15 end root plus square root of 15 cross times 8 end root minus square root of 8 cross times 15 end root minus square root of 8 cross times 8 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 75 plus square root of 40 over denominator square root of 15 squared end root plus square root of 120 minus square root of 120 minus square root of 64 end fraction end cell end table

Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction end cell equals cell fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction cross times fraction numerator square root of 15 plus square root of 8 over denominator square root of 15 plus square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 plus square root of 5 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses over denominator open parentheses square root of 15 minus square root of 8 close parentheses open parentheses square root of 15 plus square root of 8 close parentheses end fraction end cell row blank equals cell fraction numerator 4 cross times square root of 15 plus 4 cross times square root of 8 plus square root of 5 cross times square root of 15 plus square root of 5 cross times square root of 8 over denominator square root of 15 cross times square root of 15 plus square root of 15 cross times square root of 8 minus square root of 8 cross times square root of 15 minus square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 5 cross times 15 end root plus square root of 5 cross times 8 end root over denominator square root of 15 cross times 15 end root plus square root of 15 cross times 8 end root minus square root of 8 cross times 15 end root minus square root of 8 cross times 8 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 75 plus square root of 40 over denominator square root of 15 squared end root plus square root of 120 minus square root of 120 minus square root of 64 end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 25 cross times 3 end root plus square root of 4 cross times 10 end root over denominator 15 minus 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus square root of 5 squared cross times 3 end root plus square root of 2 squared cross times 10 end root over denominator 7 end fraction end cell row blank equals cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus 5 square root of 3 plus 2 square root of 10 over denominator 7 end fraction end cell end table

Jadi bentuk rasional dari fraction numerator 4 plus square root of 5 over denominator square root of 15 minus square root of 8 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 4 square root of 15 plus 4 square root of 8 plus 5 square root of 3 plus 2 square root of 10 over denominator 7 end fraction end cell end table.

h. Diketahui fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction maka cara merasionalkannya adalah dengan mengalikan penyebut dengan akar sekawannya yaitu 4 square root of 3 minus square root of 6 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction end cell equals cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction cross times fraction numerator 4 square root of 3 minus square root of 6 over denominator 4 square root of 3 minus square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 8 minus square root of 2 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses over denominator open parentheses 4 square root of 3 plus square root of 6 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 8 cross times 4 square root of 3 minus square root of 8 cross times square root of 6 minus square root of 2 cross times 4 square root of 3 plus square root of 2 cross times square root of 6 over denominator 4 square root of 3 cross times 4 square root of 3 minus 4 square root of 3 cross times square root of 6 plus square root of 6 cross times 4 square root of 3 minus square root of 6 cross times square root of 6 end fraction end cell end table

Berdasarkan aturan perkalian bentuk akar dengan akar pangkat sama yaitu n-th root of a cross times n-th root of b equals n-th root of a cross times b end root maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction end cell equals cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction cross times fraction numerator 4 square root of 3 minus square root of 6 over denominator 4 square root of 3 minus square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 8 minus square root of 2 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses over denominator open parentheses 4 square root of 3 plus square root of 6 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 8 cross times 4 square root of 3 minus square root of 8 cross times square root of 6 minus square root of 2 cross times 4 square root of 3 plus square root of 2 cross times square root of 6 over denominator 4 square root of 3 cross times 4 square root of 3 minus 4 square root of 3 cross times square root of 6 plus square root of 6 cross times 4 square root of 3 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 8 cross times 3 end root minus square root of 8 cross times 6 end root minus 4 square root of 2 cross times 3 end root plus square root of 2 cross times 6 end root over denominator 16 square root of 3 cross times 3 end root minus 4 square root of 3 cross times 6 end root plus 4 square root of 6 cross times 3 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 24 minus square root of 48 minus 4 square root of 6 plus square root of 12 over denominator 16 square root of 9 minus 4 square root of 18 plus 4 square root of 18 minus square root of 36 end fraction end cell end table

Jika n-th root of a dapat difaktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q bilangan berpangkat n maka n-th root of a dapat disederhanakan sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction end cell equals cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction cross times fraction numerator 4 square root of 3 minus square root of 6 over denominator 4 square root of 3 minus square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 8 minus square root of 2 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses over denominator open parentheses 4 square root of 3 plus square root of 6 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 8 cross times 4 square root of 3 minus square root of 8 cross times square root of 6 minus square root of 2 cross times 4 square root of 3 plus square root of 2 cross times square root of 6 over denominator 4 square root of 3 cross times 4 square root of 3 minus 4 square root of 3 cross times square root of 6 plus square root of 6 cross times 4 square root of 3 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 8 cross times 3 end root minus square root of 8 cross times 6 end root minus 4 square root of 2 cross times 3 end root plus square root of 2 cross times 6 end root over denominator 16 square root of 3 cross times 3 end root minus 4 square root of 3 cross times 6 end root plus 4 square root of 6 cross times 3 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 24 minus square root of 48 minus 4 square root of 6 plus square root of 12 over denominator 16 square root of 9 minus 4 square root of 18 plus 4 square root of 18 minus square root of 36 end fraction end cell row blank equals cell fraction numerator 4 square root of 4 cross times 6 end root minus square root of 16 cross times 3 end root minus 4 square root of 6 plus square root of 4 cross times 3 end root over denominator 16 cross times 9 minus 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 2 squared cross times 6 end root minus square root of 4 squared cross times 3 end root minus 4 square root of 6 plus square root of 2 squared cross times 3 end root over denominator 144 minus 6 end fraction end cell row blank equals cell fraction numerator 4 cross times 2 square root of 6 minus 4 square root of 3 minus 4 square root of 6 plus 2 square root of 3 over denominator 138 end fraction end cell row blank equals cell fraction numerator 8 square root of 6 minus 4 square root of 3 minus 4 square root of 6 plus 2 square root of 3 over denominator 138 end fraction end cell end table

Berdasarkan aturan penjumlahan dan pengurangan bentuk akar dengan bilangan pokok sama yaitu a n-th root of b plus-or-minus c n-th root of b equals open parentheses a plus-or-minus c close parentheses n-th root of b maka

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction end cell equals cell fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction cross times fraction numerator 4 square root of 3 minus square root of 6 over denominator 4 square root of 3 minus square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 8 minus square root of 2 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses over denominator open parentheses 4 square root of 3 plus square root of 6 close parentheses open parentheses 4 square root of 3 minus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 8 cross times 4 square root of 3 minus square root of 8 cross times square root of 6 minus square root of 2 cross times 4 square root of 3 plus square root of 2 cross times square root of 6 over denominator 4 square root of 3 cross times 4 square root of 3 minus 4 square root of 3 cross times square root of 6 plus square root of 6 cross times 4 square root of 3 minus square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 8 cross times 3 end root minus square root of 8 cross times 6 end root minus 4 square root of 2 cross times 3 end root plus square root of 2 cross times 6 end root over denominator 16 square root of 3 cross times 3 end root minus 4 square root of 3 cross times 6 end root plus 4 square root of 6 cross times 3 end root minus square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 24 minus square root of 48 minus 4 square root of 6 plus square root of 12 over denominator 16 square root of 9 minus 4 square root of 18 plus 4 square root of 18 minus square root of 36 end fraction end cell row blank equals cell fraction numerator 4 square root of 4 cross times 6 end root minus square root of 16 cross times 3 end root minus 4 square root of 6 plus square root of 4 cross times 3 end root over denominator 16 cross times 9 minus 6 end fraction end cell row blank equals cell fraction numerator 4 square root of 2 squared cross times 6 end root minus square root of 4 squared cross times 3 end root minus 4 square root of 6 plus square root of 2 squared cross times 3 end root over denominator 144 minus 6 end fraction end cell row blank equals cell fraction numerator 4 cross times 2 square root of 6 minus 4 square root of 3 minus 4 square root of 6 plus 2 square root of 3 over denominator 138 end fraction end cell row blank equals cell fraction numerator 8 square root of 6 minus 4 square root of 3 minus 4 square root of 6 plus 2 square root of 3 over denominator 138 end fraction end cell row blank equals cell fraction numerator 8 square root of 6 minus 4 square root of 6 minus 4 square root of 3 plus 2 square root of 3 over denominator 138 end fraction end cell row blank equals cell fraction numerator open parentheses 8 minus 4 close parentheses square root of 6 minus open parentheses 4 minus 2 close parentheses square root of 3 over denominator 138 end fraction end cell row blank equals cell fraction numerator 4 square root of 6 minus 2 square root of 3 over denominator 138 end fraction end cell end table

Jadi bentuk rasional dari fraction numerator square root of 8 minus square root of 2 over denominator 4 square root of 3 plus square root of 6 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 4 square root of 6 minus 2 square root of 3 over denominator 138 end fraction end cell end table.

0

Roboguru

Sederhanakan bentuk akar berikut. d.

Pembahasan Soal:

Dengan menggunakan sifat begin mathsize 14px style square root of a b end root equals square root of a cross times square root of b end style diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 48 end cell equals cell square root of 16 cross times 3 end root end cell row blank equals cell square root of 16 cross times square root of 3 end cell row blank equals cell 4 cross times square root of 3 end cell row blank equals cell 4 square root of 3 end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 192 end cell equals cell square root of 64 cross times 3 end root end cell row blank equals cell square root of 64 cross times square root of 3 end cell row blank equals cell 8 cross times square root of 3 end cell row blank equals cell 8 square root of 3 end cell end table end style

Selanjutnya, dengan menggunakan sifat penjumlahan dan pengurangan bentuk akar dengan bilangan pokok sama

begin mathsize 14px style a square root of c plus-or-minus b square root of c equals open parentheses a plus-or-minus b close parentheses square root of c end style

diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 7 square root of 3 plus square root of 48 minus square root of 192 end cell equals cell 7 square root of 3 plus 4 square root of 3 minus 8 square root of 3 end cell row blank equals cell open parentheses 7 plus 4 minus 8 close parentheses square root of 3 end cell row blank equals cell 3 square root of 3 end cell end table end style

Jadi, bentuk sederhana dari begin mathsize 14px style 7 square root of 3 plus square root of 48 minus square root of 192 end style adalah begin mathsize 14px style 3 square root of 3 end style.

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