Răspuns: Ai rezolvarea mai jos
Explicație pas cu pas:
Buna!
Punctul a)
[tex]\bf a = \bigg\{ \bigg[\Big(2^{4} \Big)^{3}: \Big(2^{2} \Big)^{2}\cdot\Big(2^{6} \Big)^{2}\bigg]^{2} :\Big(2^{8} \Big)^{4}\bigg\}^{6}:\Big(-2^{23} \Big)^{2}[/tex]
[tex]\bf a =\bigg[\Big(2^{4\cdot3}: 2^{2\cdot 2}\cdot2^{6\cdot 2}\Big)^{2} :2^{8\cdot 4}\bigg]^{6}:\Big(2^{23} \Big)^{2}[/tex]
[tex]\bf a =\bigg[\Big(2^{12}: 2^{4}\cdot2^{12}\Big)^{2} :2^{32}\bigg]^{6}:2^{23\cdot2}[/tex]
[tex]\bf a =\bigg[\Big(2^{12-4+12}\Big)^{2} :2^{32}\bigg]^{6}:2^{46}[/tex]
[tex]\bf a =\bigg[\Big(2^{20}\Big)^{2} :2^{32}\bigg]^{6}:2^{46}[/tex]
[tex]\bf a =\bigg(2^{20\cdot 2} :2^{32}\bigg)^{6}:2^{46}[/tex]
[tex]\bf a =\Big(2^{40-32} \Big)^{6}:2^{46}[/tex]
[tex]\bf a =\Big(2^{8} \Big)^{6}:2^{46}[/tex]
[tex]\bf a =2^{8\cdot 6}:2^{46}[/tex]
[tex]\bf a =2^{48}:2^{46}[/tex]
[tex]\bf a =2^{48-46}[/tex]
[tex]\pink{\bf \boxed{\bf a =2^{2}\longrightarrow Patrat~perfect}}[/tex]
La b îmi iese un număr f mare si anume: 3¹⁹⁴/2¹⁹², dar il scriu așa
[tex]\bf b = \bigg\{ \bigg[\bigg(\dfrac{3}{2} \bigg)^{3}\bigg]^{10}: \bigg[\bigg(-\dfrac{2}{3} \bigg)^{4}\bigg]^{6}\cdot \bigg(\dfrac{9}{4} \bigg)^{2}\bigg\}^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 4[/tex]
[tex]\bf b = \bigg\{ \bigg[\bigg(\dfrac{3}{2} \bigg)^{3\cdot 10}: \bigg[\bigg(\dfrac{2}{3} \bigg)^{4}\bigg]^{6}\cdot \bigg[\bigg(\dfrac{3}{2}\bigg)^{2} \bigg]^{2}\bigg\}^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg[\bigg(\dfrac{3}{2} \bigg)^{30}: \bigg(\dfrac{2}{3} \bigg)^{4\cdot 6}\cdot \bigg(\dfrac{3}{2}\bigg)^{2\cdot 2}\bigg]^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg[\bigg(\dfrac{3}{2} \bigg)^{30}: \bigg(\dfrac{2}{3} \bigg)^{24}\cdot \bigg(\dfrac{3}{2}\bigg)^{4}\bigg]^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg[\bigg(\dfrac{3}{2} \bigg)^{30}: \bigg(\dfrac{3}{2} \bigg)^{-24}\cdot \bigg(\dfrac{3}{2}\bigg)^{4}\bigg]^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg[\bigg(\dfrac{3}{2} \bigg)^{30-(-24)+4}~\bigg]^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg[\bigg(\dfrac{3}{2} \bigg)^{30+24+4}~\bigg]^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg[\bigg(\dfrac{3}{2} \bigg)^{58}~\bigg]^{4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg(\dfrac{3}{2} \bigg)^{58\cdot4}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b = \bigg(\dfrac{3}{2} \bigg)^{232}:\bigg(\dfrac{3}{2} \bigg)^{38}\cdot 2^{2}[/tex]
[tex]\bf b =\dfrac{3^{232}}{2^{232}}~:~\dfrac{3^{38}}{2^{38}} \cdot 2^{2}[/tex]
[tex]\bf b =\dfrac{\not3^{232}}{\not2^{232}}~\cdot~\dfrac{\not2^{38}}{\not3^{38}}~ \cdot ~\not2^{2}[/tex]
[tex]\bf b =\dfrac{3^{232-38}}{2^{232-38-2}}[/tex]
[tex]\bf b =\dfrac{3^{194}}{2^{192}}[/tex]
[tex]\bf b =\dfrac{\Big(3^{97}\Big)^{2}}{\Big(2^{96}\Big)^{2}}[/tex]
[tex]\red{\boxed{\bf b =\Bigg(\dfrac{3^{97}}{2^{96}}\Bigg)^{2}\longrightarrow Patrat~ perfect}}[/tex]
Punctul b)
[tex]\bf\purple{ \bf a^{b} = 4^{\bigg(\dfrac{3^{97}}{2^{96}}\bigg)^{2}} \implies Patrat ~perfect}[/tex]
[tex]\bf \green{ b^{a} = \bigg[ \bigg(\dfrac{3^{97}}{2^{96}}\bigg)^{2}\bigg]^{4} \implies Patrat ~perfect}[/tex]
Bafta multa!
