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Pertanyaan

Rasionalkan penyebut pecahan-pecahan berikut!
a. fraction numerator 14 a over denominator square root of 7 a end root end fraction
b. fraction numerator square root of 5 over denominator 6 square root of 15 end fraction
c. fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction
d. fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction

R. Febrianti

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Merasionalkan bentuk akar adalah dengan cara mengalikan penyebut akar dengan akar sekawannya sehingga jika fraction numerator a over denominator square root of b end fraction maka 

fraction numerator a over denominator square root of b end fraction equals fraction numerator a over denominator square root of b end fraction cross times fraction numerator square root of b over denominator square root of b end fraction

Sehingga

a. fraction numerator 14 a over denominator square root of 7 a end root end fraction diperoleh sebagai berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 14 a over denominator square root of 7 a end root end fraction end cell equals cell fraction numerator 14 a over denominator square root of 7 a end root end fraction cross times fraction numerator square root of 7 a end root over denominator square root of 7 a end root end fraction end cell row blank equals cell fraction numerator 14 a cross times square root of 7 a end root over denominator square root of 7 a end root cross times square root of 7 a end root 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 sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 14 a over denominator square root of 7 a end root end fraction end cell equals cell fraction numerator 14 a over denominator square root of 7 a end root end fraction cross times fraction numerator square root of 7 a end root over denominator square root of 7 a end root end fraction end cell row blank equals cell fraction numerator 14 a cross times square root of 7 a end root over denominator square root of 7 a end root cross times square root of 7 a end root end fraction end cell row blank equals cell fraction numerator 14 a cross times square root of 7 a end root over denominator square root of 7 a cross times 7 a end root end fraction end cell row blank equals cell fraction numerator 14 a cross times square root of 7 a end root over denominator square root of 49 a squared end root end fraction end cell row blank equals cell fraction numerator begin display style stack up diagonal strike 14 a end strike with 2 on top end style square root of 7 a end root over denominator up diagonal strike 7 a end strike end fraction end cell row blank equals cell 2 square root of 7 a end root end cell end table

Jadi bentuk rasional dari fraction numerator 14 a over denominator square root of 7 a end root end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 square root of 7 a end root end cell end table.

b. fraction numerator square root of 5 over denominator 6 square root of 15 end fraction diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 5 over denominator 6 square root of 15 end fraction end cell equals cell fraction numerator square root of 5 over denominator 6 square root of 15 end fraction cross times fraction numerator square root of 15 over denominator square root of 15 end fraction end cell row blank equals cell fraction numerator square root of 5 cross times square root of 15 over denominator 6 square root of 15 cross times square root of 15 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 sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 5 over denominator 6 square root of 15 end fraction end cell equals cell fraction numerator square root of 5 over denominator 6 square root of 15 end fraction cross times fraction numerator square root of 15 over denominator square root of 15 end fraction end cell row blank equals cell fraction numerator square root of 5 cross times square root of 15 over denominator 6 square root of 15 cross times square root of 15 end fraction end cell row blank equals cell fraction numerator square root of 5 cross times 15 end root over denominator 6 square root of 15 cross times 15 end root end fraction end cell row blank equals cell fraction numerator square root of 75 over denominator 6 square root of 15 squared end root 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 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator square root of 5 over denominator 6 square root of 15 end fraction end cell equals cell fraction numerator square root of 5 over denominator 6 square root of 15 end fraction cross times fraction numerator square root of 15 over denominator square root of 15 end fraction end cell row blank equals cell fraction numerator square root of 5 cross times square root of 15 over denominator 6 square root of 15 cross times square root of 15 end fraction end cell row blank equals cell fraction numerator square root of 5 cross times 15 end root over denominator 6 square root of 15 cross times 15 end root end fraction end cell row blank equals cell fraction numerator square root of 75 over denominator 6 square root of 15 squared end root end fraction end cell row blank equals cell fraction numerator square root of 25 cross times 3 end root over denominator 6 cross times 15 end fraction end cell row blank equals cell fraction numerator square root of 5 squared cross times 3 end root over denominator 6 cross times 15 end fraction end cell row blank equals cell fraction numerator up diagonal strike 5 square root of 3 over denominator 6 cross times begin display style stack up diagonal strike 15 with 3 below end style end fraction end cell row blank equals cell fraction numerator square root of 3 over denominator 18 end fraction end cell end table

Jadi bentuk rasional dari fraction numerator square root of 5 over denominator 6 square root of 15 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator square root of 3 over denominator 18 end fraction end cell end table.

c. fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction diperoleh sebagai berikut.

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

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction end cell equals cell fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction cross times fraction numerator square root of 6 over denominator square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses 3 square root of 3 plus 7 close parentheses square root of 6 over denominator square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 cross times square root of 6 plus 7 cross times square root of 6 over denominator square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 cross times 6 end root plus 7 square root of 6 over denominator square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 3 square root of 18 plus 7 square root of 6 over denominator square root of 6 squared end root 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 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction end cell equals cell fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction cross times fraction numerator square root of 6 over denominator square root of 6 end fraction end cell row blank equals cell fraction numerator open parentheses 3 square root of 3 plus 7 close parentheses square root of 6 over denominator square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 cross times square root of 6 plus 7 cross times square root of 6 over denominator square root of 6 cross times square root of 6 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 cross times 6 end root plus 7 square root of 6 over denominator square root of 6 cross times 6 end root end fraction end cell row blank equals cell fraction numerator 3 square root of 18 plus 7 square root of 6 over denominator square root of 6 squared end root end fraction end cell row blank equals cell fraction numerator 3 square root of 9 cross times 2 end root plus 7 square root of 6 over denominator 6 end fraction end cell row blank equals cell fraction numerator 3 square root of 3 squared cross times 2 end root plus 7 square root of 6 over denominator 6 end fraction end cell row blank equals cell fraction numerator 3 cross times 3 square root of 2 plus 7 square root of 6 over denominator 6 end fraction end cell row blank equals cell fraction numerator 9 square root of 2 plus 7 square root of 6 over denominator 6 end fraction end cell end table

Jadi bentuk rasional dari fraction numerator 3 square root of 3 plus 7 over denominator square root of 6 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 9 square root of 2 plus 7 square root of 6 over denominator 6 end fraction end cell end table.

d. fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction end cell equals cell fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction cross times fraction numerator square root of 8 over denominator square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 square root of 5 minus square root of 6 close parentheses cross times square root of 8 over denominator 3 square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 cross times square root of 8 minus square root of 6 cross times square root of 8 over denominator 3 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 sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction end cell equals cell fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction cross times fraction numerator square root of 8 over denominator square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 square root of 5 minus square root of 6 close parentheses cross times square root of 8 over denominator 3 square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 cross times square root of 8 minus square root of 6 cross times square root of 8 over denominator 3 square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 cross times 8 end root minus square root of 6 cross times 8 end root over denominator 3 square root of 8 cross times 8 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 40 minus square root of 48 over denominator 3 square root of 8 squared end root 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 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction end cell equals cell fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction cross times fraction numerator square root of 8 over denominator square root of 8 end fraction end cell row blank equals cell fraction numerator open parentheses 4 square root of 5 minus square root of 6 close parentheses cross times square root of 8 over denominator 3 square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 cross times square root of 8 minus square root of 6 cross times square root of 8 over denominator 3 square root of 8 cross times square root of 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 cross times 8 end root minus square root of 6 cross times 8 end root over denominator 3 square root of 8 cross times 8 end root end fraction end cell row blank equals cell fraction numerator 4 square root of 40 minus square root of 48 over denominator 3 square root of 8 squared end root end fraction end cell row blank equals cell fraction numerator 4 square root of 4 cross times 10 end root minus square root of 16 cross times 3 end root over denominator 3 cross times 8 end fraction end cell row blank equals cell fraction numerator 4 square root of 2 squared cross times 10 end root minus square root of 4 squared cross times 3 end root over denominator 24 end fraction end cell row blank equals cell fraction numerator 4 cross times 2 square root of 10 minus 4 square root of 3 over denominator 24 end fraction end cell row blank equals cell fraction numerator up diagonal strike 4 open parentheses 2 square root of 10 minus square root of 3 close parentheses over denominator up diagonal strike 4 cross times 6 end fraction end cell row blank equals cell fraction numerator 2 square root of 10 minus square root of 3 over denominator 6 end fraction end cell end table

Jadi bentuk rasional dari fraction numerator 4 square root of 5 minus square root of 6 over denominator 3 square root of 8 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 square root of 10 minus square root of 3 over denominator 6 end fraction end cell end table.

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