a friend shared this challenging problem to me. 2^y ×y^2 +(2y)^(2y) =272 no inspection approach! thank you all.


Question and Answers Forum

All Questions Topic List
None Questions
Previous in All Question Next in All Question
Previous in None Next in None
Question Number 188284 by Humble last updated on 27/Feb/23

$a f r i e n d s h a r e d t h i s c h a l l e n g i n g$ $p r o b l e m t o m e .$ $2^{y} \times y^{2} + {(2 y)}^{2 y} = 272$ $n o i n s p e c t i o n a p p r o a c h!$ $t h a n k y o u a l l .$
Commented by Frix last updated on 27/Feb/23

$I {don}^{'} t think {there}^{'} s a method for an exact$ $solution . You can approximate or use$ $your brains . I showed that$ $2^{y} \times y^{2} + {(2 y)}^{2 y} = 16 + 256$ $\Rightarrow y = 2$ $If you {don}^{'} t like this approach, find another$ $one!$
	Answered by a.lgnaoui last updated on 28/Feb/23
	$Authere methode Remarque: 272=2^4 +2^8 2^y ×y^2 +2^(2y) ×y^(2y) =272 (2^y )×y^2 +(2^y )^2 ×(y^2 )^y =2^4 +2^8 =2^4 +(2^4 )^2 { (((2^y )×y^2 = a)),(((2^y )^2 ×(y^y )^2 =b)) :} (b/a)=2^y ×y^(2y−2) =2^y ×y^(2(y−1)) a+b=2^y y^2 +2^y ×y^2 (2^y ×y^(2(y−1)) ) =y^2 2^y +2^(2y) y^(2y) =2^y y^2 (1+2^y y^(2(y−1)) ) =2^y y^2 (1+(2^(2y) /4))=2^y y^2 (1+(((2^y )^2 )/4)) a+b=272=2^4 +2^8 a+b= 2^4 (1+(1/(16)) ) { ((y^2 2^y =2^4 (1))),((((2^(2y) /4))=(2^4 /4^2 ) (2))) :} y=2$
	$A u t h e r e m e t h o d e$ $R e m a r q u e : 272 = 2^{4} + 2^{8}$ $2^{y} \times y^{2} + 2^{2 y} \times y^{2 y} = 272$ $(2^{y}) \times y^{2} + {(2^{y})}^{2} \times {(y^{2})}^{y} = 2^{4} + 2^{8}$ $= 2^{4} + {(2^{4})}^{2}$ ${\begin{cases} (2^{y}) \times y^{2} = a \\ {(2^{y})}^{2} \times {(y^{y})}^{2} = b \end{cases}$ $\frac{b}{a} = 2^{y} \times y^{2 y - 2} = 2^{y} \times y^{2 (y - 1)}$ $a + b = 2^{y} y^{2} + 2^{y} \times y^{2} (2^{y} \times y^{2 (y - 1)})$ $= y^{2} 2^{y} + 2^{2 y} y^{2 y} = 2^{y} y^{2} (1 + 2^{y} y^{2 (y - 1)})$ $= 2^{y} y^{2} (1 + \frac{2^{2 y}}{4}) = 2^{y} y^{2} (1 + \frac{{(2^{y})}^{2}}{4})$ $a + b = 272 = 2^{4} + 2^{8}$ $a + b = 2^{4} (1 + \frac{1}{16})$ ${\begin{cases} y^{2} 2^{y} = 2^{4} (1) \\ (\frac{2^{2 y}}{4}) = \frac{2^{4}}{4^{2}} (2) \end{cases}$ $y = 2$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum