Question

Brain Teaser #13

Title: Math_Master

$\\\\\textbf{There are \underline{two solutions}. Can you find both of them?}}\\\\\text{\underline{Solve:}}\\\\\huge{x^{\sqrt{x}}} = x\sqrt x$

P.S - Your answer must have explanation. It should not be ridiculous.

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Answer 1

x√x=x^3/2

Integration x^32=x^(3/2+1)÷3/2+1

=X^5/2÷5/2

=(2/5)x^5/2

Answer 2

Answer:

$x^{\sqrt{x}}=x\sqrt{x}\\\\\implies x^{\sqrt{x}}=x^1\times x^{1/2}\\\\\implies x^{\sqrt{x}}=x^{1+1/2}\\\\\implies x^{\sqrt{x}}=x^{3/2}\rightarrow (1)\\\\\implies x^{\sqrt{x}}-x^{3/2}=0\\\\\implies x(x^{\sqrt{x}-1}-x^{3/2-1})=0\\\\\therefore x\neq 0 [\textbf{Reciprocal of zero does not exist]}\\\\\implies x=1[\textbf{By trial and error]}\\\\\textsf{From (1) we have :}\\\\\implies x^{\sqrt{x}}=x^{3/2}\\\\\implies \sqrt{x}=\dfrac{3}{2}\\\\\implies x=\dfrac{9}{4}$

x can be 1 or x can be 9/4 .

The two solutions are 1 and 9/4 .