Aptitude Question ID : 93836

If x^{\sqrt[x]{x}} = \left ( x\sqrt{x} \right )^{x}, then x equals :
[A]\frac{3}{2}
[B]\frac{9}{4}
[C]\frac{4}{9}
[D]\frac{2}{3}

\mathbf{\frac{9}{4}}
Given Expression,
x^{\sqrt[x]{x}} = \left ( x\sqrt{x} \right )^{x}
=> x^{x.x^{\frac{1}{2}}} = \left ( x\times x^{\frac{1}{2}}\right )^{x}
=> x^{x^{1+\frac{1}{2}}} = \left ( x^{1+\frac{1}{2}} \right )^{x}
=> x^{x^{\frac{3}{2}}} = \left ( x\frac{3}{2} \right )^{x} = x^{\frac{3x}{2}}
=> x^{\frac{3x}{2}} = \frac{3x}{2}= x^{\frac{3x}{2}} - \frac{3x}{2} = 0
=> x\left ( x^{\frac{1}{2}} - \frac{3}{2} \right ) = 0
=> x = 0
or x^{\frac{1}{2}} = \frac{3}{2}
=> x = \left ( \frac{3}{2} \right )^{2} = \frac{9}{4}
x = 0 given indeterminate value.
\therefore x = \frac{9}{4}
Hence option [B] is the right answer.

Advertisement

Comments