Roboguru

Sederhanakan 7​+6​1​ dan 7​−6​1​, kemudian tentukan hasil operasi berikut! a. 7​+6​1​+7​−6​1​, b. (7​+6​1​)2 dan(7​−6​1​)2, c. (7​+6​1​−7​−6​1​)2.

Pertanyaan

Sederhanakan fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction dan fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction, kemudian tentukan hasil operasi berikut!
a. fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction,
b. open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared text  dan end text space open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared,
c. open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared

Pembahasan Soal:

Diketahui fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction maka bentuk sederhana dari fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction adalah dengan merasionalkan fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction dengan cara mengalikan fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction dengan akar sekawannya yaitu square root of 7 minus square root of 6 sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell fraction numerator 1 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 square root of 7 minus square root of 6 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 square root of 7 minus 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 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell fraction numerator 1 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 square root of 7 minus square root of 6 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 square root of 7 minus 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 square root of 7 minus square root of 6 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 square root of 7 minus square root of 6 over denominator square root of 49 minus square root of 42 plus square root of 42 minus square root of 36 end fraction end cell row blank equals cell fraction numerator square root of 7 minus square root of 6 over denominator 7 minus 6 end fraction end cell row blank equals cell fraction numerator square root of 7 minus square root of 6 over denominator 1 end fraction end cell row blank equals cell square root of 7 minus square root of 6 end cell end table

Diketahui fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction maka bentuk sederhana dari fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction adalah dengan merasionalkan fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction dengan cara mengalikan fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction dengan akar sekawannya yaitu square root of 7 plus square root of 6 sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction end cell equals cell fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction cross times fraction numerator square root of 7 plus square root of 6 over denominator square root of 7 plus square root of 6 end fraction end cell row blank equals cell fraction numerator square root of 7 plus square root of 6 over denominator open parentheses square root of 7 minus square root of 6 close parentheses open parentheses square root of 7 plus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 7 plus square root of 6 over denominator square root of 7 cross times square root of 7 plus square root of 7 cross times square root of 6 minus 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 diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction end cell equals cell fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction cross times fraction numerator square root of 7 plus square root of 6 over denominator square root of 7 plus square root of 6 end fraction end cell row blank equals cell fraction numerator square root of 7 plus square root of 6 over denominator open parentheses square root of 7 minus square root of 6 close parentheses open parentheses square root of 7 plus square root of 6 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 7 plus square root of 6 over denominator square root of 7 cross times square root of 7 plus square root of 7 cross times square root of 6 minus 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 square root of 7 plus square root of 6 over denominator square root of 7 cross times 7 end root plus square root of 7 cross times 6 end root minus 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 square root of 7 plus square root of 6 over denominator square root of 49 plus square root of 42 minus square root of 42 minus square root of 36 end fraction end cell row blank equals cell fraction numerator square root of 7 plus square root of 6 over denominator 7 minus 6 end fraction end cell row blank equals cell fraction numerator square root of 7 plus square root of 6 over denominator 1 end fraction end cell row blank equals cell square root of 7 plus square root of 6 end cell end table

Berdasarkan hitungan di atas diperoleh bahwa table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction end cell equals cell square root of 7 minus square root of 6 end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction end cell equals cell square root of 7 plus square root of 6 end cell end table sehingga

a. fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction diperoleh sebagai berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction end cell equals cell square root of 7 minus square root of 6 plus square root of 7 plus square root of 6 end cell row blank equals cell square root of 7 plus square root of 7 up diagonal strike negative square root of 6 plus square root of 6 end strike end cell row blank equals cell open parentheses 1 plus 1 close parentheses square root of 7 end cell row blank equals cell 2 square root of 7 end cell end table

Jadi hasil dari fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction plus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction adalah 2 square root of 7.

b. open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared diperoleh sebagai berikut

open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared equals open parentheses square root of 7 minus square root of 6 close parentheses squared

Berdasarkan rumus pengkuadratan suku dua bentuk aljabar yaitu 

open parentheses a minus b close parentheses squared equals a squared minus 2 a b plus b squared

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared 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

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times 6 end root plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 square root of 42 plus open parentheses square root of 6 close parentheses squared end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat yaitu n-th root of a to the power of m end root equals a to the power of m over n end exponent comma space a not equal to 0 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times 6 end root plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 7 to the power of 1 half end exponent close parentheses squared minus 2 square root of 42 plus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell end table

Berdasarkan sifat bilangan berpangkat open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponent dan a to the power of 1 equals a maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 7 cross times 6 end root plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 cross times square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared minus 2 square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 7 to the power of 1 half end exponent close parentheses squared minus 2 square root of 42 plus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 7 to the power of 1 half cross times 2 end exponent minus 2 square root of 42 plus 6 to the power of 1 half cross times 2 end exponent end cell row blank equals cell 7 to the power of 1 minus 2 square root of 42 plus 6 to the power of 1 end cell row blank equals cell 7 minus 2 square root of 42 plus 6 end cell row blank equals cell 13 minus 2 square root of 42 end cell end table

Dan open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared diperoleh sebagai berikut

open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared equals open parentheses square root of 7 plus square root of 6 close parentheses squared

Berdasarkan rumus pengkuadratan suku dua bentuk aljabar yaitu 

open parentheses a plus b close parentheses squared equals a squared plus 2 a b plus b squared

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 plus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared 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

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 plus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times 6 end root plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 square root of 42 plus open parentheses square root of 6 close parentheses squared end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat yaitu n-th root of a to the power of m end root equals a to the power of m over n end exponent comma space a not equal to 0 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 plus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times 6 end root plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 7 to the power of 1 half end exponent close parentheses squared plus 2 square root of 42 plus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell end table

Berdasarkan sifat bilangan berpangkat open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponent dan a to the power of 1 equals a maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 plus square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times square root of 6 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 7 cross times 6 end root plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 cross times square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses square root of 7 close parentheses squared plus 2 square root of 42 plus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 7 to the power of 1 half end exponent close parentheses squared plus 2 square root of 42 plus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 7 to the power of 1 half cross times 2 end exponent plus 2 square root of 42 plus 6 to the power of 1 half cross times 2 end exponent end cell row blank equals cell 7 to the power of 1 plus 2 square root of 42 plus 6 to the power of 1 end cell row blank equals cell 7 plus 2 square root of 42 plus 6 end cell row blank equals cell 13 plus 2 square root of 42 end cell end table

Jadi hasil dari open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction close parentheses squared equals 13 minus 2 square root of 42 dan open parentheses fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared equals 13 plus 2 square root of 42.

c. open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared diperoleh sebagai berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses square root of 7 minus square root of 6 minus open parentheses square root of 7 plus square root of 6 close parentheses close parentheses squared end cell row blank equals cell open parentheses square root of 7 minus square root of 6 minus square root of 7 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses up diagonal strike square root of 7 minus square root of 7 end strike minus square root of 6 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses negative square root of 6 minus square root of 6 close parentheses squared end cell row blank equals cell open parentheses open parentheses negative 1 minus 1 close parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses negative 2 square root of 6 close parentheses squared end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat yaitu n-th root of a to the power of m end root equals a to the power of m over n end exponent comma space a not equal to 0 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses negative 2 square root of 6 close parentheses squared end cell row blank equals cell open parentheses negative 2 cross times 6 to the power of 1 half end exponent close parentheses squared end cell end table

Berdasarkan sifat bilangan berpangkat open parentheses a cross times b close parentheses to the power of m equals a to the power of m cross times b to the power of m maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses negative 2 square root of 6 close parentheses squared end cell row blank equals cell open parentheses negative 2 cross times 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell open parentheses negative 2 close parentheses squared cross times open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell end table

Berdasarkan sifat bilangan berpangkat yaitu open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponent dan a to the power of 1 equals a serta bilangan negatif berpangkat genap yaitu open parentheses negative a close parentheses to the power of n equals a to the power of n comma space n space text bilangan genap end text maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared end cell equals cell open parentheses negative 2 square root of 6 close parentheses squared end cell row blank equals cell open parentheses negative 2 cross times 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell open parentheses negative 2 close parentheses squared cross times open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 4 cross times 6 to the power of 1 half cross times 2 end exponent end cell row blank equals cell 4 cross times 6 to the power of 1 end cell row blank equals cell 4 cross times 6 end cell row blank equals 24 end table

Jadi hasil dari open parentheses fraction numerator 1 over denominator square root of 7 plus square root of 6 end fraction minus fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction close parentheses squared adalah 24.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Febrianti

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 07 Oktober 2021

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