Question

Fie numerele:
a=$\sqrt{5+2 \sqrt{6}}$
b=$\sqrt{5-2 \sqrt{6}}$
a)Calculati media aritmetica a numerelor a si b
b)Calculati media geometrica a numerelor a si b

Answer 1

a= $\sqrt{( \sqrt{3}+ \sqrt{2} ) ^{2} }$= I$\sqrt{3} + \sqrt{2}$I= $\sqrt{3} + \sqrt{2}$
b=$\sqrt{( \sqrt{3}- \sqrt{2} ) ^{2} }$= I$\sqrt{3} - \sqrt{2}$I= $\sqrt{3} - \sqrt{2}$
ma =( a+b)/2=( $\sqrt{3} + \sqrt{2}$+$\sqrt{3} - \sqrt{2}$)/2 = 2*$\sqrt{3}$/2 = $\sqrt{3}$
mg = $\sqrt{a*b}$ =( $\sqrt{3} + \sqrt{2}$)*( $\sqrt{3} - \sqrt{2}$)=3-2=1

Answer 2

a=√(5+2√6)
b=√(5-2√6)

se foloseste formula
√(A+/-√B)=√(A+C)/2  +/- √(A-C)/2    unde C=√(A²-B)
in cazul nostru
A=5
B=2²×6=24
C=√(A²-B)=√(25-24)=1

√(5+2√6)=√(5+1)/2+√(5-1)/2 =√3+√2
√(5-2√6)=√(5+1)/2-√(5-1)/2 =√3-√2


a) ma=[√(5+2√6) +√(5-2√6)]/2=(√3+√2+√3-√2)/2=√3
b) mg=√(√(5+2√6) ×√(5-2√6)]/2=√(√3+√2)(√3-√2)=√(3-2)=√1=1