Question

30 Se considera numerele:​

Answer 1

Explicație pas cu pas:

a) cu o modificare de enunț, la numărul b

$a = {\Big\{ {\Big[ {( {2}^{4} )}^{3} : {( {2}^{2} )}^{2} \cdot {( {2}^{6} )}^{2} \Big] }^{2} : {( {2}^{8} )}^{4} \Big\}}^{6} : {( - {2}^{23} )}^{2}=$

$= {\Big[ {\Big({2}^{12} : {2}^{4} \cdot {2}^{12} \Big) }^{2} : {2}^{32} \Big]}^{6} : {2}^{46} = {\Big[ {\Big({2}^{12 - 4 + 12} \Big) }^{2} : {2}^{32} \Big]}^{6} : {2}^{46}$

$= {\Big[ {\Big({2}^{20} \Big) }^{2} : {2}^{32} \Big]}^{6} : {2}^{46} = {\Big( {2}^{40} : {2}^{32} \Big)}^{6} : {2}^{46}$

$= {\Big( {2}^{8} \Big)}^{6} : {2}^{46} = {2}^{48} : {2}^{46} = {2}^{2} = \bf 4$

$b = {\Big\{ {\Big[ {\Big( \frac{3}{2} \Big)}^{3} \Big] }^{10} : {\Big[ {\Big( \red{ \bf - \frac{3}{2}} \Big)}^{4} \Big] }^{6} \cdot {\Big( \frac{9}{4} \Big)}^{2}\Big\}}^{4} : {\Big( \frac{3}{2} \Big)}^{38} \cdot 4 =$

$= {\Big[ { \Big( \frac{3}{2} \Big)}^{30} : {\Big(\frac{3}{2} \Big)}^{24} \cdot {\Big( \frac{ {3}^{2} }{ {2}^{2} } \Big)}^{2}\Big]}^{4} : {\Big( \frac{3}{2} \Big)}^{38} \cdot 4$

$= {\Big \{ { \Big( \frac{3}{2} \Big)}^{30 - 24} \cdot {\Big[ {\Big( \frac{3}{2} \Big)}^{2} \Big] }^{2}\Big \}}^{4} : {\Big( \frac{3}{2} \Big)}^{38} \cdot 4$

$= {\Big[ { \Big( \frac{3}{2} \Big)}^{6} \cdot { \Big( \frac{3}{2} \Big)}^{4}\Big]}^{4} : {\Big( \frac{3}{2} \Big)}^{38} \cdot 4$

$= {\Big[ { \Big( \frac{3}{2} \Big)}^{10} \Big]}^{4} : {\Big( \frac{3}{2} \Big)}^{38} \cdot 4 = { \Big( \frac{3}{2} \Big)}^{40} : {\Big( \frac{3}{2} \Big)}^{38} \cdot 4$

$= {\Big( \frac{3}{2} \Big)}^{2} \cdot 4 = \frac{9}{4} \cdot 4 = \bf 9$

b)

${a}^{b} = {4}^{9} = {( {2}^{2} )}^{9} = \bf {( {2}^{9} )}^{2} = {512}^{2}$

${b}^{a} = {9}^{4} = {9}^{2 \cdot 2} = \bf {( {9}^{2} )}^{2} = {81}^{2}$

q.e.d.