Roboguru

Panjang sebuah persegi panjang  kali lebarnya. Jika luas persegi panjang tersebut , tentukan keliling dan panjang diagonalnya dalam bentuk akar yang sederhana!

Pertanyaan

Panjang sebuah persegi panjang 4 kali lebarnya. Jika luas persegi panjang tersebut 72 space text cm end text squared, tentukan keliling dan panjang diagonalnya dalam bentuk akar yang sederhana!

Pembahasan Soal:

Diketahui:

  • Misal lebar persegi panjang: l
  • Panjang persegi panjang open parentheses p close parentheses4 space text kali lebarnya end text equals 4 l
  • Luas persegi panjang open parentheses text L end text close parentheses72 space text cm end text squared

Ditanya:

  1. Kelilingnya
  2. Panjang diagonalnya

Ingat kembali rumus luas persegi panjang yaitu

text L end text equals p cross times l

Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell text L end text end cell equals cell p cross times l end cell row 72 equals cell 4 l cross times l end cell row 72 equals cell 4 l squared end cell row cell 72 over 4 end cell equals cell l squared end cell row 18 equals cell l squared end cell row cell plus-or-minus square root of 18 space text cm end text end cell equals l end table

Karena l merupakan ukuran suatu bangun datar maka nilainya tidak mungkin negatif sehingga l equals square root of 18 space text cm end text.

Ingat jika n-th root of a dapat di faktorkan menjadi n-th root of p cross times q end root dengan p space text atau end text space q merupakan bilangan pangkat n maka n-th root of a dapat disederhanakan.

Sehingga square root of 18 diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 18 end cell equals cell square root of 9 cross times 2 end root end cell row blank equals cell square root of 3 squared cross times 2 end root end cell row blank equals cell 3 square root of 2 end cell end table

Sehingga l equals 3 square root of 2 space text cm end text. Panjang persegi panjang 4 kali lebarnya maka 

table attributes columnalign right center left columnspacing 0px end attributes row p equals cell 4 l end cell row blank equals cell 4 cross times 3 square root of 2 end cell row blank equals cell 12 square root of 2 space text cm end text end cell end table

Ingat rumus keliling persegi panjang yaitu

text K end text equals 2 open parentheses p plus l close parentheses

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell text K end text end cell equals cell 2 open parentheses p plus l close parentheses end cell row blank equals cell 2 open parentheses 12 square root of 2 plus 3 square root of 2 close parentheses end cell row blank equals cell 2 open parentheses open parentheses 12 plus 3 close parentheses square root of 2 close parentheses end cell row blank equals cell 2 open parentheses 15 square root of 2 close parentheses end cell row blank equals cell 30 square root of 2 space text cm end text end cell end table

Berdasarkan karakteristik persegi panjang yaitu keempat sudutnya siku-siku dan diagonalnya membagi persegi panjang menjadi dua bagian sama besar seperti ilustrasi gambar berikut.


Berdasarkan gambar di atas maka terbentuk dua segitiga siku-siku sama besar sehingga panjang diagonal persegi panjang open parentheses d close parentheses dapat ditentukan menggunakan teorema pythagoras sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row d equals cell square root of p squared plus l squared end root end cell row blank equals cell square root of open parentheses 12 square root of 2 close parentheses squared plus open parentheses 3 square root of 2 close parentheses squared end root end cell end table

Berdasarkan hubungan akar bilangan dengan pangkat pecahan yaitu n-th root of a to the power of m end root equals a to the power of m over n end exponent dengan n not equal to 0 maka

table attributes columnalign right center left columnspacing 0px end attributes row d equals cell square root of p squared plus l squared end root end cell row blank equals cell square root of open parentheses 12 square root of 2 close parentheses squared plus open parentheses 3 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 12 cross times 2 to the power of 1 half end exponent close parentheses squared plus open parentheses 3 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell end table

Berdasarkan sifat bilangan berpangkat yaitu 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 d equals cell square root of p squared plus l squared end root end cell row blank equals cell square root of open parentheses 12 square root of 2 close parentheses squared plus open parentheses 3 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 12 cross times 2 to the power of 1 half end exponent close parentheses squared plus open parentheses 3 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 12 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared plus 3 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared end root 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 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row d equals cell square root of p squared plus l squared end root end cell row blank equals cell square root of open parentheses 12 square root of 2 close parentheses squared plus open parentheses 3 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 12 cross times 2 to the power of 1 half end exponent close parentheses squared plus open parentheses 3 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 12 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared plus 3 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 144 cross times 2 to the power of 1 half cross times 2 end exponent plus 9 cross times 2 to the power of 1 half cross times 2 end exponent end root end cell row blank equals cell square root of 144 cross times 2 to the power of 1 plus 9 cross times 2 to the power of 1 end root end cell row blank equals cell square root of 144 cross times 2 plus 9 cross times 2 end root end cell row blank equals cell square root of 288 plus 18 end root end cell row blank equals cell square root of 306 end cell row blank equals cell square root of 9 cross times 34 end root end cell row blank equals cell square root of 3 squared cross times 34 end root end cell row blank equals cell 3 square root of 34 space text cm end text end cell end table

Jadi berdasarkan hitungan di atas didapatkan bahwa keliling persegi panjang dan panjang diagonalnya masing-masing adalah 30 square root of 2 space text cm dan end text space 3 square root of 34 space text cm end text.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 13 Agustus 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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

Pembahasan Soal:

Ingat sifat distributif perkalian sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell a open parentheses b plus c close parentheses end cell equals cell a b plus b c end cell row cell open parentheses a plus b close parentheses open parentheses c plus d close parentheses end cell equals cell a c plus a d plus b c plus b d end cell end table

Sehingga

d. square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses end cell equals cell square root of 3 cross times 2 square root of 6 minus square root of 3 cross times square root of 25 end cell row blank equals cell square root of 3 cross times 2 square root of 6 minus square root of 3 cross times 5 end cell row blank blank blank 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 comma a not equal to 0 space text dan end text space b not equal to 0 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses end cell equals cell square root of 3 cross times 2 square root of 6 minus square root of 3 cross times square root of 25 end cell row blank equals cell square root of 3 cross times 2 square root of 6 minus square root of 3 cross times 5 end cell row blank equals cell 2 square root of 3 cross times 6 end root minus 5 square root of 3 end cell row blank equals cell 2 square root of 18 minus 5 square root of 3 end cell row blank blank blank 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 square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses end cell equals cell 2 square root of 3 cross times 6 end root minus 5 square root of 3 end cell row blank equals cell 2 square root of 18 minus 5 square root of 3 end cell row blank equals cell 2 square root of 9 cross times 2 end root minus 5 square root of 3 end cell row blank equals cell 2 square root of 3 squared cross times 2 end root minus 5 square root of 3 end cell row blank equals cell 2 cross times 3 square root of 2 minus 5 square root of 3 end cell row blank equals cell 6 square root of 2 minus 5 square root of 3 end cell end table

Jadi hasil dari square root of 3 open parentheses 2 square root of 6 minus square root of 25 close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 6 square root of 2 minus 5 square root of 3 end cell end table.

e. open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses Diperoleh sebagai berikut.

begin mathsize 12px style open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses equals square root of 6 cross times square root of 6 minus square root of 6 cross times square root of 3 plus square root of 8 cross times square root of 6 minus square root of 8 cross times square root of 3 end style

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

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses end cell equals cell square root of 6 cross times square root of 6 minus square root of 6 cross times square root of 3 plus square root of 8 cross times square root of 6 minus square root of 8 cross times square root of 3 end cell row blank equals cell square root of 6 cross times 6 end root minus square root of 6 cross times 3 end root plus square root of 8 cross times 6 end root minus square root of 8 cross times 3 end root end cell row blank equals cell square root of 36 minus square root of 18 plus square root of 48 minus square root of 24 end cell end table end style

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 open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses end cell equals cell square root of 6 cross times 6 end root minus square root of 6 cross times 3 end root plus square root of 8 cross times 6 end root minus square root of 8 cross times 3 end root end cell row blank equals cell square root of 36 minus square root of 18 plus square root of 48 minus square root of 24 end cell row blank equals cell 6 minus square root of 9 cross times 2 end root plus square root of 16 cross times 3 end root minus square root of 4 cross times 6 end root end cell row blank equals cell 6 minus square root of 3 squared cross times 3 end root plus square root of 4 squared cross times 3 end root minus square root of 2 squared cross times 6 end root end cell row blank equals cell 6 minus 3 square root of 3 plus 4 square root of 3 minus 2 square root of 6 end cell end table

Bentuk akar dengan akar dan bilangan pokok sama dapat dikurangkan dengan menggunakan sifat distributif sebagai berikut

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

Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses end cell equals cell square root of 6 cross times 6 end root minus square root of 6 cross times 3 end root plus square root of 8 cross times 6 end root minus square root of 8 cross times 3 end root end cell row blank equals cell square root of 36 minus square root of 18 plus square root of 48 minus square root of 24 end cell row blank equals cell 6 minus square root of 9 cross times 2 end root plus square root of 16 cross times 3 end root minus square root of 4 cross times 6 end root end cell row blank equals cell 6 minus square root of 3 squared cross times 3 end root plus square root of 4 squared cross times 3 end root minus square root of 2 squared cross times 6 end root end cell row blank equals cell 6 minus 3 square root of 3 plus 4 square root of 3 minus 2 square root of 6 end cell row blank equals cell 6 minus square root of 3 open parentheses 3 minus 4 close parentheses minus 2 square root of 6 end cell row blank equals cell 6 minus open parentheses negative 1 close parentheses square root of 3 minus 2 square root of 6 end cell row blank equals cell 6 plus square root of 3 minus 2 square root of 6 end cell row blank blank blank end table

Jadi hasil dari open parentheses square root of 6 plus square root of 8 close parentheses open parentheses square root of 6 minus square root of 3 close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 6 plus square root of 3 minus 2 square root of 6 end cell row blank blank blank end table.

f. open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses diperoleh sebagai berikut.

begin mathsize 12px style open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses equals 2 square root of 3 cross times square root of 8 minus 2 square root of 3 cross times square root of 2 minus square root of 6 cross times square root of 8 plus square root of 6 cross times square root of 2 end style

Berdasarkan 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

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses end cell equals cell 2 square root of 3 cross times square root of 8 minus 2 square root of 3 cross times square root of 2 minus square root of 6 cross times square root of 8 plus square root of 6 cross times square root of 2 end cell row blank equals cell 2 square root of 3 cross times 8 end root minus 2 square root of 3 cross times 2 end root minus square root of 6 cross times 8 end root plus square root of 6 cross times 2 end root end cell row blank equals cell 2 square root of 24 minus 2 square root of 6 minus square root of 48 plus square root of 12 end cell row blank blank blank end table end style

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 open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses end cell equals cell 2 square root of 3 cross times 8 end root minus 2 square root of 3 cross times 2 end root minus square root of 6 cross times 8 end root plus square root of 6 cross times 2 end root end cell row blank equals cell 2 square root of 24 minus 2 square root of 6 minus square root of 48 plus square root of 12 end cell row blank equals cell 2 square root of 4 cross times 6 end root minus 2 square root of 6 minus square root of 16 cross times 3 end root plus square root of 4 cross times 3 end root end cell row blank equals cell 2 square root of 2 squared cross times 6 end root minus 2 square root of 6 minus square root of 4 squared cross times 3 end root plus square root of 2 squared cross times 3 end root end cell row blank equals cell 2 cross times 2 square root of 6 minus 2 square root of 6 minus 4 square root of 3 plus 2 square root of 3 end cell row blank equals cell 4 square root of 6 minus 2 square root of 6 minus 4 square root of 3 plus 2 square root of 3 end cell end table

Bentuk akar dengan akar dan bilangan pokok sama dapat dikurangkan dengan menggunakan sifat distributif sebagai berikut

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

Sehingga

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

Jadi hasil dari open parentheses 2 square root of 3 minus square root of 6 close parentheses open parentheses square root of 8 minus square root of 2 close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 square root of 6 minus 2 square root of 3 end cell end table.

g. Ingat rumus perkalian suku dua bentuk aljabar yaitu

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

Sehingga open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses diperoleh sebagai berikut.

open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses equals open parentheses 3 square root of 5 close parentheses squared minus open parentheses square root of 6 close parentheses squared

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 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses end cell equals cell open parentheses 3 square root of 5 close parentheses squared minus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 3 cross times 5 to the power of 1 half end exponent close parentheses squared minus 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 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 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses end cell equals cell open parentheses 3 square root of 5 close parentheses squared minus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 3 cross times 5 to the power of 1 half end exponent close parentheses squared minus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 3 squared cross times open parentheses 5 to the power of 1 half end exponent close parentheses squared minus 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 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses end cell equals cell open parentheses 3 square root of 5 close parentheses squared minus open parentheses square root of 6 close parentheses squared end cell row blank equals cell open parentheses 3 cross times 5 to the power of 1 half end exponent close parentheses squared minus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 3 squared cross times open parentheses 5 to the power of 1 half end exponent close parentheses squared minus open parentheses 6 to the power of 1 half end exponent close parentheses squared end cell row blank equals cell 9 cross times 5 to the power of 1 half cross times 2 end exponent minus 6 to the power of 1 half cross times 2 end exponent end cell row blank equals cell 9 cross times 5 to the power of 1 minus 6 to the power of 1 end cell row blank equals cell 9 cross times 5 minus 6 end cell row blank equals cell 45 minus 6 end cell row blank equals 39 end table

Jadi hasil dari open parentheses 3 square root of 5 minus square root of 6 close parentheses open parentheses 3 square root of 5 plus square root of 6 close parentheses adalah 39.

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Roboguru

pada gambar di atas adalah layang-layang. Diagonalnya berpotongan di titik . Jika panjang , hitunglah: b. Keliling layang-layang .

Pembahasan Soal:

Diketahui layang-layang text PQRS end text dengan panjang text PR end text equals 8 square root of 6 space text cm end text comma space text dan QS end text equals 8 square root of 2 space text cm end text seperti pada gambar di bawah ini.

Diketahui text PO end text equals 3 square root of 6 space text cm dan end text space text PR end text equals 8 square root of 6 space text cm end text sehingga panjang text OR end text yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell text OR end text end cell equals cell text PR end text minus text PO end text end cell row blank equals cell 8 square root of 6 minus 3 square root of 6 end cell row blank equals cell open parentheses 8 minus 3 close parentheses square root of 6 end cell row blank equals cell 5 square root of 6 space text cm end text end cell end table

Berdasarkan karakteristik layang-layang yaitu salah satu diagonalnya membagi diagonal yang lain sama besar maka text QO end text equals text OS end text equals fraction numerator 8 square root of 2 over denominator 2 end fraction equals 4 square root of 2 space text cm end text.

Berdasarkan karakteristik layang-layang yaitu kedua diagonalnya saling berpotongan tegak lurus maka angle text POQ end text merupakan sudut siku-siku sehingga terbentuk segitiga siku-siku seperti pada gambar di bawah ini.

Sehingga panjang text PQ end text dapat ditentukan dengan menggunakan teorema pythagoras sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell text PQ end text end cell equals cell square root of text PO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 3 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat dengan aturan sebagai berikut.

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 text dengan end text space n not equal to 0

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell text PQ end text end cell equals cell square root of text PO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 3 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 3 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell end table

Berdasarkan sifat bilangan berpangkat yaitu 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 dan 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 exponentmaka

table attributes columnalign right center left columnspacing 0px end attributes row cell text PQ end text end cell equals cell square root of text PO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 3 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 3 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 3 squared cross times open parentheses 6 to the power of 1 half end exponent close parentheses squared plus 4 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 9 cross times 6 to the power of 1 half cross times 2 end exponent plus 16 cross times 2 to the power of 1 half cross times 2 end exponent end root end cell row blank equals cell square root of 9 cross times 6 to the power of 1 plus 16 cross times 2 to the power of 1 end root end cell row blank equals cell square root of 9 cross times 6 plus 16 cross times 2 end root end cell row blank equals cell square root of 54 plus 32 end root end cell row blank equals cell square root of 86 space text cm end text end cell end table

Berdasarkan karakteristik layang-layang maka text PS end text equals text PQ end text equals square root of 86 space text cm end text

Berdasarkan karakteristik layang-layang yaitu kedua diagonalnya saling berpotongan tegak lurus maka angle text ROQ end text merupakan sudut siku-siku sehingga terbentuk segitiga siku-siku seperti pada gambar di bawah ini.

Sehingga panjang text RQ end text dapat ditentukan dengan menggunakan teorema pythagoras sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell text RQ end text end cell equals cell square root of text RO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 5 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat dengan aturan sebagai berikut.

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 text dengan end text space n not equal to 0

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell text RQ end text end cell equals cell square root of text RO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 5 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 5 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell end table

Berdasarkan sifat bilangan berpangkat yaitu 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 dan 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 exponentmaka

table attributes columnalign right center left columnspacing 0px end attributes row cell text RQ end text end cell equals cell square root of text RO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 5 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 5 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 5 squared cross times open parentheses 6 to the power of 1 half end exponent close parentheses squared plus 4 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 25 cross times 6 to the power of 1 half cross times 2 end exponent plus 16 cross times 2 to the power of 1 half cross times 2 end exponent end root end cell row blank equals cell square root of 25 cross times 6 to the power of 1 plus 16 cross times 2 to the power of 1 end root end cell row blank equals cell square root of 9 cross times 6 plus 16 cross times 2 end root end cell row blank equals cell square root of 150 plus 32 end root end cell row blank equals cell square root of 182 space text cm end text end cell end table

Berdasarkan karakteristik layang-layang maka text RS end text equals text RQ end text equals square root of 182 space text cm end text. Keliling layang-layang dapat ditentukan dengan menjumlahkan semua sisi-sisinya sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell text K end text end cell equals cell text PQ end text plus text PS end text plus text RQ end text plus text RS end text end cell row blank equals cell square root of 84 plus square root of 84 plus square root of 182 plus square root of 182 end cell row blank equals cell 2 square root of 84 plus 2 square root of 182 end cell row blank equals cell 2 open parentheses square root of 84 plus square root of 182 close parentheses space text cm end text end cell end table

Jadi keliling layang-layang tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 open parentheses square root of 84 plus square root of 182 close parentheses space text cm end text end cell end table.

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Hasil pemangkatan dari  adalah

Pembahasan Soal:

Ingat n-th root of a to the power of m end root equals open parentheses a to the power of m close parentheses to the power of n comma space a not equal to 0 maka square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root diperoleh sebagai berikut.

square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root equals open parentheses m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent close parentheses to the power of 1 half end exponent

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 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root close parentheses to the power of 4 end cell equals cell open parentheses open parentheses m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent close parentheses to the power of 1 half end exponent close parentheses to the power of 4 end cell row blank equals cell open parentheses m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent close parentheses to the power of 1 half cross times 4 end exponent end cell row blank equals cell open parentheses m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent close parentheses squared end cell end table

Pecahan campuran 1 3 over 4 dapat diubah menjadi pecahan senilai sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 3 over 4 end cell equals cell fraction numerator 1 cross times 4 plus 3 over denominator 4 end fraction end cell row blank equals cell fraction numerator 4 plus 3 over denominator 4 end fraction end cell row blank equals cell 7 over 4 end cell end table

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root close parentheses to the power of 4 end cell equals cell open parentheses m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent close parentheses squared end cell row blank equals cell open parentheses m to the power of 2 over 3 end exponent n to the power of 7 over 4 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 diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root close parentheses to the power of 4 end cell equals cell open parentheses m to the power of 2 over 3 end exponent n to the power of 7 over 4 end exponent close parentheses squared end cell row blank equals cell open parentheses m to the power of 2 over 3 end exponent close parentheses squared open parentheses n to the power of 7 over 4 end exponent close parentheses squared end cell row blank equals cell m to the power of 2 over 3 cross times 2 end exponent n to the power of fraction numerator 7 over denominator begin display style stack up diagonal strike 4 with 2 below end style end fraction cross times up diagonal strike 2 end exponent end cell row blank equals cell m to the power of 4 over 3 end exponent n to the power of 7 over 2 end exponent end cell end table

Pecahan 4 over 3 space text dan end text space 7 over 2 dapat dijadikan pecahan dengan penyebut sama yaitu berpenyebut KPK dari 3 space text dan end text space 2 yaitu 6 maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 over 3 end cell equals cell fraction numerator 4 cross times 2 over denominator 3 cross times 2 end fraction end cell row blank equals cell 8 over 6 end cell row cell 7 over 2 end cell equals cell fraction numerator 7 cross times 3 over denominator 2 cross times 3 end fraction end cell row blank equals cell 21 over 6 end cell end table

Berdasarkan sifat bilangan berpangkat pecahan yaitu a to the power of m over n end exponent equals n-th root of a to the power of m end root maka diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root close parentheses to the power of 4 end cell equals cell m to the power of 4 over 3 end exponent n to the power of 7 over 2 end exponent end cell row blank equals cell m to the power of 8 over 6 end exponent n to the power of 21 over 6 end exponent end cell row blank equals cell m to the power of 8 cross times 1 over 6 end exponent n to the power of 21 cross times 1 over 6 end exponent end cell row blank equals cell open parentheses m to the power of 8 n to the power of 21 close parentheses to the power of 1 over 6 end exponent end cell row blank equals cell root index 6 of m to the power of 8 n to the power of 21 end root 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 open parentheses square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root close parentheses to the power of 4 end cell equals cell root index 6 of m to the power of 8 n to the power of 21 end root end cell row blank equals cell root index 6 of m to the power of 6 cross times m squared cross times n to the power of 18 cross times n cubed end root end cell row blank equals cell m to the power of 6 over 6 end exponent n to the power of 18 over 6 end exponent root index 6 of m squared n cubed end root end cell row blank equals cell m to the power of 1 n cubed root index 6 of m squared n cubed end root end cell row blank equals cell m n cubed root index 6 of m squared n cubed end root end cell end table

Jadi hasil pemangkatan dari open parentheses square root of m to the power of 2 over 3 end exponent n to the power of 1 3 over 4 end exponent end root close parentheses to the power of 4 adalah m n cubed space root index 6 of m squared n cubed end root.

Oleh karena itu, jawaban yang benar adalah A.

3

Roboguru

Pembahasan Soal:

Ingat bahwa:

(am)n=am×nnam=anm 

Sehingga penyelesaiannya adalah sebagai berikut:

684×3813=========6(23)4×3(92)36(2)3×4×3(9)2×36212×3962612×392622×393222×93321×912×918 

Jadi, jawaban yang tepat adalah D.

1

Roboguru

Tentukan hasil perkalian akar-akar bilangan berikut! e.

Pembahasan Soal:

Ingat kembali rumus berikut.

  • 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 
  • n-th root of a to the power of m end root equals a to the power of m over n end exponent 
  • a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent 
  • square root of a squared cross times b end root equals a square root of b
  • open parentheses square root of a minus square root of b close parentheses open parentheses square root of a plus square root of b close parentheses equals open parentheses square root of a close parentheses squared minus open parentheses square root of b close parentheses squared 

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 2 fourth root of a to the power of 5 end root plus square root of b close parentheses open parentheses 2 fourth root of a to the power of 5 end root minus square root of b close parentheses end cell equals cell 2 squared open parentheses fourth root of a to the power of 5 end root close parentheses squared minus open parentheses square root of b close parentheses squared end cell row blank equals cell 4 open parentheses a to the power of 5 over 4 end exponent close parentheses squared minus b end cell row blank equals cell 4 open parentheses a to the power of 5 over 4 cross times 2 end exponent close parentheses minus b end cell row blank equals cell 4 open parentheses a to the power of 5 over 2 end exponent close parentheses minus b end cell row blank equals cell 4 open parentheses a to the power of 2 1 half end exponent close parentheses minus b end cell row blank equals cell 4 open parentheses a to the power of 2 plus 1 half end exponent close parentheses minus b end cell row blank equals cell 4 open parentheses a squared cross times a to the power of 1 half end exponent close parentheses minus b end cell row blank equals cell 4 a squared square root of a minus b end cell end table 

Jadi, hasil dari open parentheses 2 fourth root of a to the power of 5 end root plus square root of b close parentheses open parentheses 2 fourth root of a to the power of 5 end root minus square root of b close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table squared table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of a end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative b end cell end table.

0

Roboguru

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