Question

Calculati media aritmetică și media geometrica a numerelor :
$a = \sqrt{(5 + 3 \sqrt{3} } ) ^{2}$
$b = |5 - 3 \sqrt{3} |$
Va rog frumos ​

Answer 1

a=√(5+3√3)²

a=|5+3√3|

a=5+3√3

b=|5-3√3|

b=3√3-5

ma=a+b/2

ma=5+3√3+3√3-5/2

ma=6√3/2

ma=3√3

mg=√a×b

mg=√(5+3√3)(3√3-5)

mg=√(3√3+5)(3√3-5)

mg=√(3√3)²-5²

mg=√27-25

mg=√2

Answer 2

$\it a = \sqrt{(5 + 3\sqrt{3})^{2} }  <=> a =  |5 + 3\sqrt{3}| <=> a = 5 + 3\sqrt{3} \\ \\ b =  |5 - 3\sqrt{3}| = -(5 - 3\sqrt{3}) = -5 + 3\sqrt{3}  \\ \\ m_{a} = \frac{a~+~b}{2} = \frac{5 + 3\sqrt{3} - 5 + 3\sqrt{3} }{2} = \frac{6\sqrt{3} }{2} = 3\sqrt{3}  \\ \\ m_{g} = \sqrt{a \cdot b} = \sqrt{(5 + 3\sqrt{3})(-5 + 3\sqrt{3})} = \sqrt{(3\sqrt{3} + 5)(3\sqrt{3} - 5) }  = \sqrt{(3\sqrt{3})^{2} - 5^{2}  } = \\ = \sqrt{27 - 25} = \sqrt{2}$