Köklü İfadenin Üslü İfade Şekilde Yazılması

$$\Large \sqrt[n]{a^m } = a^{\frac{m}{n} }$$

Örnek:

$\bullet \large\sqrt[4]{ 8} = \sqrt[4]{2^3 } = 2^{ \frac{3}{4} } $

$\bullet \large\sqrt[3]{ -2} = (-2)^{ \frac{1}{3} } $

Soru 2

$ \large \sqrt[3]{2^x } = \sqrt{(0,5)^{2x-1} } $ ise x kaçtır?

\[ \text{A)} 3 \quad \text{B) } 2 \quad \text{C) } 1 \quad \text{D) } \frac{3}{8} \quad \text{E)} \frac{1}{6} \]

Çözüm:

\[ \large\sqrt[3]{2^x} = \sqrt{(0,5)^{2x-1}} \]
\[ \large \Rightarrow 2^{\Large\frac{x}{3}} = \left( \frac{1}{2} \right)^{\Large\frac{2x-1}{2}} \]
\[ \large \Rightarrow 2^{\Large\frac{x}{3}} = (2^{-1})^{\Large\frac{2x-1}{2}} \]
\[ \large \Rightarrow 2^{\Large\frac{x}{3}} = 2^{\Large\frac{-2x+1}{2}} \]
\[ \Rightarrow \frac{x}{3} = \frac{1 – 2x}{2} \] \[ \Rightarrow x = \frac{3}{8} \quad \text{dir.} \]

$\textbf{Cevab: D} $

Soru 3

\[ \large\sqrt[4]{\left( \frac{3}{20} \right)^{3x+1}} = \frac{400}{9} \] ise x kaçtır?

\[ \text{A)} -4 \quad \text{B) } -3 \quad \text{C) } 0 \quad \text{D) } 3 \quad \text{E)} 4 \]

Çözüm:

\[ \sqrt[4]{\left( \frac{3}{20} \right)^{3x+1}} = \frac{400}{9} \]

\[ \left( \frac{3}{20} \right)^{\large\frac{3x+1}{4}} = \left( \frac{20}{3}\right)^2 = \left( \frac{3}{20}\right)^{-2} \]

\[ \Rightarrow \frac{3x +1 }{4} = -2 \]

\[ \Rightarrow 3x +1 = -8 \]

\[ \Rightarrow 3x = -9 \]

\[ \Rightarrow x= -3 \]

$\textbf{Cevab: B} $