Question

a si b numere reale, a la puterea 2 + b la a 2=4 AoriB=1. caluclat:
(a+b)la a 2
(a-b)la a 2
a/b+b/a
a la a 4 + b la 4

Answer 1

a² + b² =4•a•b = 1

1) (a+b)² = a² + b² + 2ab = 4ab + 2ab = 1 + 1/2 = 3/2;

2) (a - b)² = a² + b² - 2ab = 1 - 1/2 = 1/2;

3) 4ab = 1 => b = 1/4a => a/b = a/ (1/4a) = a•4a/1 = 4a²

=> b/a = 1/4a²
 
=> a/b + b/a = 4a² + 1/4a² = (16a⁴ + 1)/ 4a²;

Adunăm ce am obținut la 1) şi 2) => a²+ b² + a² - b² = 3/2 + 1/2 => 2a² = 4/2 = 2

=> a² =1 => a/b + b/a = (16•1² +1)/ 4•1 = 17/4

4) a² + b² = 1 |² => a⁴ + b⁴ + 2(ab)² = 1 => a⁴ + b⁴ = 1 - 2•(1/4)² = 1 - 2•1/16 = 1 - 1/8 = 7/8