Fie nr nat x= 20x5la puterea 2015 x2 la puterea 2014 -2017 aratati ca x+2017 este patrat perfect si cub perfect

x = 20 × 5^2015 × 2^2014 - 2017

x = 2^2 × 5 × 5^2015 × 2^2014 - 2017

x = 2^2 × 2^2014 × 5 × 5^2015 - 2017

x = 2^2016 × 5^2016 - 2017

x = (2 × 5)^2016 - 2017

x = 10^2016 - 2017

x + 2017 = 10^2016 - 2017 + 2017 = 10^2016

10^2016 = (10^1008)^2 este p.p
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10^2016 = (10^672)^3 este cub perfect
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Răspuns

(10¹⁰⁰⁸)² => p.p

(10⁶⁷²)³ => c.p

Explicație pas cu pas:

x=20*5²⁰¹⁵ * 2²⁰¹⁴ - 2017 = 2² *5¹ *5²⁰¹⁵ * 2²⁰¹⁴ - 2017 = 5⁽²⁰¹⁵⁺¹⁾ *2⁽²⁰¹⁴⁺²⁾ -2017 =5²⁰¹⁶ *2²⁰¹⁶ -2017 =(5*2)²⁰¹⁶ -2017= 10²⁰¹⁶ - 2017

x=10²⁰¹⁶ - 2017   =>   x+2017=10²⁰¹⁶

10²⁰¹⁶=(10¹⁰⁰⁸)²=(10⁶⁷²)³   => p.p   si   c.p