3^x +4^x =5^x find x BY ISHOLA


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Question Number 52091 by Olalekan99 last updated on 03/Jan/19

$3^{x} + 4^{x} = 5^{x}$ $f i n d x$ $B Y I S H O L A$
Commented by scientist last updated on 03/Jan/19

$u s i n g b i n o m i a l e x p a n s i o n$ ${(1 + 2)}^{x} + {(1 + 3)}^{x} = {(1 + 4)}^{x}$
	Answered by scientist last updated on 03/Jan/19

	$x = 2$
	Answered by mr W last updated on 03/Jan/19

	$f (x) = 3^{x} + 4^{x} i s i n c r e a s i n g f u n c t i o n$ $g (x) = 5^{x} i s i n c r e a s i n g f u n c t i o n$ $f (1) = 3 + 4 = 7$ $g (1) = 5$ $f (1) > g (1)$ $f (3) = 3^{3} + 4^{3} = 91$ $g (3) = 5^{3} = 125$ $f (3) < g (3)$ $\Rightarrow t h e r e i s o n e a n d o n l y o n e s o l u t i o n b e t w e e n$ $1 a n d 3 f o r f (x) = g (x) .$ $s i n c e f (2) = 3^{2} + 4^{2} = 25 = 5^{2} = g (x),$ $t h e o n l y s o l u t i o n i s x = 2 .$
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