Prove 2 <mrow class="MJX-TeXAtom-ORD"> 135 </mrow> </msup> +

Carley Haley

Carley Haley

Answered question

2022-05-07

Prove 2 135 + 3 133 < 4 108

Answer & Explanation

Erika Ayers

Erika Ayers

Beginner2022-05-08Added 12 answers

First,
2 135 = 2 7 2 128 = 2 7 ( 2 8 ) 16 < ( 2 8 ) 17 = 256 17 .
Next,
3 133 = 3 3 3 130 = 3 3 ( 3 5 ) 26 = 3 3 ( 243 ) 26 < 3 3 ( 256 ) 26 = 27 256 26 .
Thus,
2 135 + 3 133 < 256 17 + 27 256 26 = ( 1 + 26 256 9 ) 256 17 < 256 10 256 17 = 256 27 = 4 108 .
terrasson81sgt

terrasson81sgt

Beginner2022-05-09Added 2 answers

Note that 2 6 < 3 4 , so
2 135 = 2 129 2 6 < 2 129 3 4 < 3 129 3 4 = 3 133 .
Therefore
2 135 + 3 133 < 3 133 + 3 133 = 2 3 133 < 3 3 133 = 3 134 .
Next note that 3 5 < 2 8 , so
3 134 < 2 134 ( 8 / 5 ) < 2 215 < 2 216 = 4 108 .

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