Question

if x^√x=[√x]^x then x is​

Answer 1

(√x. √x )^√x = (√x)^x

[(√x)^√x]^√x = √x^x

(√x)^√x. √x = √x^x

√x. √x = √x. (bases are same therefore equating the powers.)

√x = 1

x = 1. (squaring both sides)

Answer 2

$\mathfrak{\huge{Answer:}}$

$\tt{x^{\sqrt{x}} = \sqrt{x}^x}$

We can write $\tt{\sqrt{x}^x = x^{\frac{x}{2}}}$ Because, we know that : $\tt{\sqrt{x} = x^{\frac{1}{2}}}$. Doing the same, we get :-

$\tt{x^{\sqrt{x}} = x^{\frac{x}{2}}}$

Now, as the bases are same, we can keep the powers equal to each other and then, equate their values :-

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

Do the square on both the sides. This will be done to remove the square root sign from the L.H.S. . After squaring, we get the values as :-

$\tt{x = \frac{x^{2}}{2}}$

Cancel out one x and then get the values :-

$\sf{x = 4}$

Thus, the value of x will be = $\bf{4}$