Bilangan Berpangkat Dan Bentuk Akar Kelas 9 – BAB 1

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Perpangkatan bilangan

Contoh : 

  1. -2^4 = -(2x2x2x2) = -16
  2. (-2)^4 = - 2x - 2x - 2x - 2 = 16

Sifat – sifat berlaku pada bilangan berpangkat

Contoh :

  1. 5000^0 = 1
  2. Sederhanakan bentuk dari 2^5 x 2^4 ! 
    2^5 x 2^4 = 2^{5+4} = 2^9
  3. Sederhanakan bentuk dari 2^3 : 2^1 ! 
    2^3 : 2^1 = 2^{3-1} = 2^2
  4. Sederhanakan bentuk dari \left ( 2^2 \right )^3 ! 
    \left ( 2^2 \right )^3 = \left ( 2 \right )^{2x3} = 2^6
  5. Nilai dari (2b)^3 ! 
    (2b)^3 = 2^3 x b^3 = 8b^3
  6. Tentukan nilai dari \left ( \frac{3}{5} \right )^3% ! 
    \left ( \frac{3}{5} \right )^3 = \frac{3^3}{5^3} = \frac{27}{125}
  7. Tentukan nilai dari 5^-2 ! 
    5^{-2} = \frac{1}{5}^2 = \frac{1}{25}

Bentuk baku bilangan

Contoh :

  1. 1200000 = 1,2 x 10^6
  2. 0,000056 = 5,6 x 10^-5
  3. 4356,78 = 4,35678 x 10^3

Persamaan pangkat sederhana

Contoh :

3^{x-1} = 27
3^{x-1} = 3^3
x – 1 = 3
x = 3 + 1
x = 4

Sifat – sifat bentuk akar

Contoh :

  1. 2\left ( \sqrt{3 - 5} \right ) = … 
    = \left ( 2. \sqrt{3} \right ) + \left ( 2. (-5) \right)
= \mathbf{2 \sqrt3 - 10}
  2. \left ( \sqrt{5 + 7} \right ) \left ( \sqrt{5 + 9} \right) = …. 
    = \left( \sqrt{5}. \sqrt{5} \right) + \left( \sqrt{5}. 9 \right) + \left(7.\sqrt{5} \right) + (7.9)
= \mathbf{5 + 9 \sqrt{5} + 7\sqrt{5} + 63}
= \mathbf{5 + 63 + 9 \sqrt{5} + 7\sqrt{5}}
= \mathbf{68 + 16 \sqrt{5}}
  3.  \left( 2\sqrt{4 - 5}^2 \right) = … 
    = \left( 2\sqrt{3 - 5} \right)^2
= \mathbf{\left( 2\sqrt{3 - 5}\right)\left( 2\sqrt{3 - 5}\right)}
= \mathbf{\left( 2\sqrt{3.} 2 \sqrt{3} \right) + \left 2\sqrt{3.} (- 5)\right) + \left((- 5).2\sqrt{3}\right) + (- 5. - 5))}
= \mathbf{12 - 10 \sqrt{3} - 10 \sqrt{3} + 25}
= \mathbf{12 + 25 - 10 \sqrt{3} - 10 \sqrt{3}}
= \mathbf{37 - 20 \sqrt{3}}
  4. 10\sqrt{3} : 2\sqrt{3} = 5

Merasionalkan Penyebut Pecahan Bentuk Akar

\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} x \frac{b}{\sqrt{b}} = \frac{a\sqrt{b}}{b}}
\frac{a}{b+\sqrt{c}} = \frac{a}{b+\sqrt{c}} x \frac{b-\sqrt{c}}{b-\sqrt{c}} = \frac{a(b-\sqrt{c})}{b^2 - c}
\frac{a}{b-\sqrt{c}} = \frac{a}{b-\sqrt{c}} x \frac{b + \sqrt{c}}{b + \sqrt{c}} = \frac{a(b+\sqrt{c})}{b^2-c}
\frac{a}{b-\sqrt{c}} = \frac{a}{\sqrt{b}-\sqrt{c}} x \frac{\sqrt{b} + \sqrt{c}}{\sqrt{b} + \sqrt{c}} = \frac{a\sqrt{b} + a\sqrt{c})}{b-c}
\frac{a}{\sqrt{b} + \sqrt{c}} = \frac{a}{\sqrt{b} + \sqrt{c}} x \frac{\sqrt{b} - \sqrt{c}}{\sqrt{b} - \sqrt{c}} = \frac{a\sqrt{b}- a\sqrt{c})}{b-c}

Contoh :

Rasionalkan penyebut pecahan berikut:

  1. \frac{2}{\sqrt{5}} = … 
    = \frac{2}{\sqrt{5}} x \frac{\sqrt{5}} {\sqrt{5}}

= \mathbf{\frac{2\sqrt{5}}{5}}
  2. \frac{4}{\sqrt{5} - \sqrt{3}} = … 
    = \frac{4}{\sqrt{5} - \sqrt{3}} x \frac{\sqrt{5} + \sqrt{3}} {\sqrt {5} + \sqrt{3}}

= \frac{4(\sqrt{5}+\sqrt{3})}{ {5}-{3}}

= \frac{4(\sqrt{5}+\sqrt{3})}{ {2}}

= 2(\sqrt{5}+\sqrt{3})

Menyederhanakan Bentuk \sqrt{( a + b) \pm 2\sqrt{ab}}

Contoh :

  1. \sqrt{8 + 2\sqrt{15}} = … 
    =\sqrt{(3 + 5) + 2\sqrt{3.5}}

= \sqrt{3} + \sqrt{5}
  2. \sqrt{8 - 2\sqrt{15}} = … 
    = \sqrt{(5 + 3) - 2\sqrt{3.5}}

= \sqrt{5} - \sqrt{3}