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Jawaban problem Winda

 

Jawaban problem Winda

 

Soal 1 :

Jika x+y=3 dan x^2 +y^2 =7. Nilai dari x^4 +y^4 adalah …

 

Jawaban :

(x+y)^2 =x^2 +y^2 +2xy

3^2 = 7 + 2xy

xy=1

 

(x+y)^4=x^4 +4x^3y + 6x^2y^2 + 4xy^3 + y^4

3^4 = x^4 + y^4 + 4x^2(xy) + 4y^2(xy) + 6(xy)^2

81 = x^4 + y^4 + 4x^2(1) + 4y^2(1) + 6(1)

81 = x^4 + y^4 + 4(x^2 + y^2) + 6

81 = x^4 + y^4 + 4(7) + 6

81 = x^4 + y^4 + 34

47 = x^4 + y^4

 

Soal 2

Diketahui x+ \frac{1}{x}=1

Hitung : (x+ \frac{1}{x})^2 + (x^2+ \frac{1}{x^2})^2 + (x^3+ \frac{1}{x^3})^2 + \cdot + (x^{27}+ \frac{1}{x^{27}})^2

Jawaban :

Perhatikan bahwa

(x+ \frac{1}{x})(x+ \frac{1}{x})=x^2+ \frac{1}{x^2} + 2

(1)(1)=x^2+ \frac{1}{x^2} + 2

1-2=x^2+ \frac{1}{x^2}

-1=x^2+ \frac{1}{x^2}

 

(x+ \frac{1}{x})(x^2+ \frac{1}{x^2})=x^3+ \frac{1}{x^3} + x+ \frac{1}{x}

(1)(-1)=x^3+ \frac{1}{x^3} +1

-1-(1)=x^3+ \frac{1}{x^3}

-2=x^3+ \frac{1}{x^3}

 

(x+ \frac{1}{x})(x^3+ \frac{1}{x^3})=x^4+ \frac{1}{x^4} + x^2+ \frac{1}{x^2}

(1)(-2)=x^4+ \frac{1}{x^4} -1

-2-(-1)=x^4+ \frac{1}{x^4}

-1=x^4+ \frac{1}{x^4}

 

(x+ \frac{1}{x})(x^4+ \frac{1}{x^4})=x^5+ \frac{1}{x^5} + x^3+ \frac{1}{x^3}

(1)(-1)=x^5+ \frac{1}{x^5} -2

-1-(-2)=x^5+ \frac{1}{x^5}

1=x^5+ \frac{1}{x^5}

 

Diperoleh pola

x+ \frac{1}{x}=1

x^2+ \frac{1}{x^2}=-1

x^3+ \frac{1}{x^3}=( x^2+ \frac{1}{x^2})-( x+ \frac{1}{x})=-2

x^4+ \frac{1}{x^4}=( x^3+ \frac{1}{x^3})-( x^2+ \frac{1}{x^2})=-1

x^5+ \frac{1}{x^5}=( x^4+ \frac{1}{x^4})-( x^3+ \frac{1}{x^3})=1

x^6+ \frac{1}{x^6}=( x^5+ \frac{1}{x^5})-( x^4+ \frac{1}{x^4})=2

Dan seterusnya

  

Kalau dikuadratkan tentu saja hasilnya

1+1+4+1+1+4+1+1+4+1+1+4+...+1+1+4

(sebanyak 27)

=6 \times 9=54

  

(tidak perlu ditulis semuanya, hanya untuk memperjelas)

 

Untuk Sobat Asimtot yang punya cara lebih cepat atau tips dan trik, silahkan dishare di sini ya.. Di komentar. ..

  

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  1. 12 September 2012 pukul 8:09 AM

    0,02% dari 100 milyar berapa?

    • 12 September 2012 pukul 11:29 AM

      1000000 = 1 juta
      1000000000 = 1 milyar
      100000000000 = 100 milyar

      \frac{0,02}{100} \times 100000000000= \frac{2}{10000} \times 100000000000=20000000

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