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


Răspuns :

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