First slide

Theory of equations

Question

The roots of the equation $x^{\sqrt{x}} = {(\sqrt{x})}^{x}$ are

Moderate

A

0 and 1
B

0 and 4
C

1 and 4
D

0,1 and 4

Solution

The given equation is $x^{\sqrt{x}} = {(\sqrt{x})}^{x}$
$\Rightarrow$ $x^{\sqrt{x}} = x^{x / 2}$

$\Rightarrow$ $\sqrt{x} = \frac{x}{2}$

Squaring on both sides, we get

$\Rightarrow$ $x = \frac{x^{2}}{4}$

$\Rightarrow$ $x^{2} - 4 x = 0$ $\Rightarrow x = 0, 4$ .

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