Question

Compara numerele:a= [(2*3)*5:16*3+2*10:(2*6+2*6)]•(2*13)*5 și b=[(4•3*5-3*5)•27*2+2•9*3•3*6]•2*11

Answer 1

Explicație pas cu pas:

$a = [{( {2}^{3} )}^{5} : {16}^{3} + {2}^{10} : ( {2}^{6} + {2}^{6})] \cdot {( {2}^{13} )}^{5} = [{2}^{15} : {( {2}^{4} )}^{3} + {2}^{10} : (2 \cdot {2}^{6})] \cdot {2}^{65} = ({2}^{15} : {2}^{12} + {2}^{10} : {2}^{7}) \cdot {2}^{65} = ({2}^{3} + {2}^{3}) \cdot {2}^{65} = (2 \cdot {2}^{3}) \cdot {2}^{65} = {2}^{4} \cdot {2}^{65} = {2}^{69}$

$b = [(4 \cdot {3}^{5} - {3}^{5}) \cdot {27}^{2} + 2 \cdot {9}^{3} \cdot {3}^{6}] \cdot {2}^{11} = [(3 \cdot {3}^{5}) \cdot {( {3}^{3} )}^{2} + 2 \cdot {( {3}^{2} )}^{3} \cdot {3}^{6}] \cdot {2}^{11} = ({3}^{6} \cdot {3}^{6} + 2 \cdot {3}^{6} \cdot {3}^{6}) \cdot {2}^{11} = ({3}^{12} + 2 \cdot {3}^{12}) \cdot {2}^{11} = (3 \cdot {3}^{12}) \cdot {2}^{11} = {3}^{13} \cdot {2}^{11}$

$a = {2}^{69} = {2}^{33} \cdot {2}^{36} = {( {2}^{3} )}^{11} \cdot {2}^{36} = {8}^{11} \cdot {2}^{36}$

$b = {3}^{13} \cdot {2}^{11} = {3}^{2} \cdot {(3 \cdot 2)}^{11} = {6}^{11} \cdot 9$

${8}^{11} > {6}^{11} \ \ si \ \ {2}^{36} > 9 \\ \iff {8}^{11} \cdot {2}^{36} > {6}^{11} \cdot 9 \implies a > b$