Find: x^x = 2^(√(200)) ⇒ x = ?


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Question Number 215350 by hardmath last updated on 03/Jan/25

$Find : x^{x} = 2^{\sqrt{200}} \Rightarrow x = ?$
	Answered by zetamaths last updated on 03/Jan/25

	$. i s j u s t t h e l a m b e r t f u n c t i o n$ $x^{x} = 2^{\sqrt{200}}$ $=> x = 2^{\frac{\sqrt{200}}{x}}$ $x = {(e^{l n 2})}^{\frac{\sqrt{200}}{x}}$ $\frac{\sqrt{200}}{x} l n 2 \times x = \frac{\sqrt{200}}{x} l n 2 \times (e^{\frac{l n 2 \sqrt{200}}{x}})$ $l n 2 \times \sqrt{200} = {v e}^{v}$ $W (l n 2 \times \sqrt{200}) = v$ $x = \frac{l n 2 \times \sqrt{200}}{W (l n 2 \times \sqrt{200)}}$ $b e c a u s e v = l n 2 \frac{\sqrt{200}}{x}$ $a n d l a m b e r t f u n c t i o n i s W ({x e}^{x}) = x$
	Answered by mr W last updated on 03/Jan/25

	$2^{\sqrt{200}} = 2^{10 \sqrt{2}} = {(\sqrt{2})}^{20 \sqrt{2}}$ $= {[{(\sqrt{2})}^{5}]}^{4 \sqrt{2}} = {(4 \sqrt{2})}^{4 \sqrt{2}}$ $x^{x} = 2^{\sqrt{200}} = {(4 \sqrt{2})}^{4 \sqrt{2}}$ $\Rightarrow x = 4 \sqrt{2} ✓$
	Commented by hardmath last updated on 04/Jan/25

	$thankyou dearprofessor$
	Commented by zetamaths last updated on 05/Jan/25

	$a n d m y r e s u l t ?$
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