Soal yang Akan Dibahas
Jika $ 5^{10x} = 1600 $ dan $ 2^{\sqrt{y}} = 25 $ ,
nilai $ \frac{\left(5^{x-1}\right)^5}{8^{\left(-\sqrt{y}\right)}} $
adalah ...
A). $ 50 \, $ B). $ 100 \, $ C). $ 150 \, $ D). $ 200 \, $ E). $ 250 $
A). $ 50 \, $ B). $ 100 \, $ C). $ 150 \, $ D). $ 200 \, $ E). $ 250 $
$\spadesuit $ Konsep Dasar
*). Sifat Eksponen :
1). $ x^n = y \rightarrow x = \sqrt[n]{y} $
2). $ (a^m)^n = a^{m.n} = (a^n)^m $
3). $ a^{m-n} = \frac{a^m}{a^n} $
4). $ a^{-n} = \frac{1}{a^n} $
*). Sifat Eksponen :
1). $ x^n = y \rightarrow x = \sqrt[n]{y} $
2). $ (a^m)^n = a^{m.n} = (a^n)^m $
3). $ a^{m-n} = \frac{a^m}{a^n} $
4). $ a^{-n} = \frac{1}{a^n} $
$\clubsuit $ Pembahasan
*).Mengubah yang diketahui :
$\begin{align} 5^{10x} = 1600 \rightarrow (5^{5x})^2 & = 1600 \\ 5^{5x} & = \pm \sqrt{1600} \\ 5^{5x} & = \pm 40 \\ \text{karena } 5^{5x} > 0 \, , \text{ maka } 5^{5x} & = 40 \\ 2^{\sqrt{y}} = 25 \rightarrow 2^{\sqrt{y}} & = 5^2 \end{align} $
*).Menyelesaikan soal :
$\begin{align} \frac{\left(5^{x-1}\right)^5}{8^{\left(-\sqrt{y}\right)}} & = \frac{\left(5^{5x-5}\right)}{ \frac{1} { 8^{\sqrt{y} }}} \\ & = 5^{5x-5} \times 8^{\sqrt{y}} \\ & = \frac{5^{5x}}{ 5^5} \times (2^3) ^{\sqrt{y}} \\ & = \frac{5^{5x}}{ 5^5} \times (2 ^{\sqrt{y}} )^3 \\ & = \frac{40}{ 5^5} \times (5^2)^3 \\ & = \frac{40}{ 5^5} \times 5^6 \\ & = 40 \times \frac{5^6}{5^5} \\ & = 40 \times 5^{6-5} \\ & = 40 \times 5^1 \\ & = 40 \times 5 \\ & = 200 \end{align} $
Jadi, nilai $ \frac{\left(5^{x-1}\right)^5}{8^{\left(-\sqrt{y}\right)}} = 200 . \, \heartsuit $
*).Mengubah yang diketahui :
$\begin{align} 5^{10x} = 1600 \rightarrow (5^{5x})^2 & = 1600 \\ 5^{5x} & = \pm \sqrt{1600} \\ 5^{5x} & = \pm 40 \\ \text{karena } 5^{5x} > 0 \, , \text{ maka } 5^{5x} & = 40 \\ 2^{\sqrt{y}} = 25 \rightarrow 2^{\sqrt{y}} & = 5^2 \end{align} $
*).Menyelesaikan soal :
$\begin{align} \frac{\left(5^{x-1}\right)^5}{8^{\left(-\sqrt{y}\right)}} & = \frac{\left(5^{5x-5}\right)}{ \frac{1} { 8^{\sqrt{y} }}} \\ & = 5^{5x-5} \times 8^{\sqrt{y}} \\ & = \frac{5^{5x}}{ 5^5} \times (2^3) ^{\sqrt{y}} \\ & = \frac{5^{5x}}{ 5^5} \times (2 ^{\sqrt{y}} )^3 \\ & = \frac{40}{ 5^5} \times (5^2)^3 \\ & = \frac{40}{ 5^5} \times 5^6 \\ & = 40 \times \frac{5^6}{5^5} \\ & = 40 \times 5^{6-5} \\ & = 40 \times 5^1 \\ & = 40 \times 5 \\ & = 200 \end{align} $
Jadi, nilai $ \frac{\left(5^{x-1}\right)^5}{8^{\left(-\sqrt{y}\right)}} = 200 . \, \heartsuit $