Question-7087

Question Number 7087 by Tawakalitu. last updated on 10/Aug/16

Commented by Yozzii last updated on 10/Aug/16

W = 2^{129} 3^{81} 5^{131}

X = 2^{127} 3^{81} 5^{131}

Y = 2^{126} 3^{82} 5^{131}

Z = 2^{125} 3^{82} 5^{132}

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2^{127} < 2^{129}

\Rightarrow 2^{127} 3^{81} 5^{131} < 2^{129} 3^{81} 5^{131}

X < W

W - Y = 2^{126} 3^{81} 5^{131} (2^{3} - 3) > 0 \Rightarrow W > Y

X - Y = 2^{126} 3^{81} 5^{131} (2 - 3) < 0 \Rightarrow Y > X

∴ W > Y > X

W - Z = 2^{125} 3^{81} 5^{131} (2^{4} - 3 \times 5) > 0 \Rightarrow W > Z

Z - Y = 2^{125} 3^{82} 5^{131} (5 - 2) > 0 \Rightarrow Z > Y > X

∴ X < Y < Z < W

Commented by Tawakalitu. last updated on 10/Aug/16

G r e a t, i a p p r e c i a t e . . t h a n k s s o m u c h .

Commented by Rasheed Soomro last updated on 10/Aug/16

\underline{S i m p l e a p p r o a c h}

W = 2^{129} 3^{81} 5^{131}

X = 2^{127} 3^{81} 5^{131}

Y = 2^{126} 3^{82} 5^{131}

Z = 2^{125} 3^{82} 5^{132}

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D i v i d i n g b y s a m e p o s i t i v e n u m b e r

w i l l n o t a f f e c t i n e q u a l i t y / e q u a l i t y .

S o d i v i d i n g b y 2^{125} \times 3^{81} \times 5^{131} (G C D) t o W,

X, Y a n d Z t o o b t a i n n e w q u a n t i t i e s

W^{'}, X^{^{'}}, Y^{^{'}} a n d Z^{^{'}} r e s p e c t i v e l y .

W^{'} = 2^{4} = 16

X^{'} = 2^{2} = 4

Y^{'} = 2 \times 3 = 6

Z^{'} = 3 \times 5 = 15

C l e a r l y 4 < 6 < 15 < 16

S o, X^{'} < Y^{'} < Z^{'} < W^{'}

O r X < Y < Z < W

Commented by Tawakalitu. last updated on 10/Aug/16

W o w . . i r e a l l y a p p r e c i a t e y o u r e f f o r t \dots T h a n k s s o m u c h