【數學問題】10,11,12小題+算式感謝，急，會選最佳解答?

10)
0.1^0.1 = 0.1^(1/10)
0.2^0.2 = 0.2^(2/10) = 0.04^(1/10)
0.3^0.3 = 0.3^(3/10) = 0.027^(1/10)
0.4^0.4 = 0.4^(4/10) = 0.0256^(1/10) 是最小, 選(D)。
0.5^0.5 = 0.5^(5/10) = 0.03125^(1/10)

11)
x‧3ˣ = 3^18
x = 3^(18 - x)
注意 15 < 3^(18-15) , 16 > 3^(18-16) , 故 k = 15。

12)
a = 10^(1 - √2/2) > 10^(1 - √4/2) = 10^0 = 1 , 選項(1)正確。
a = 10^(1 - √2/2) = √10^(2 - √2) > √9^(2 - √2) = √3^(4 - 2√2) > √3^(4 - 2√2.25) = √3 , 選項(2)錯誤。
b^√3 = (a^√2)^√3 = a^√6 > a^√4 = a² (因 a > 1) , 選項(3)正確。
10^0.4 = 10^(√1.96 -1) < b = a^√2 = (10^(1 - √2/2))^√2 = 10^(√2 - 1) < 10^(√2.25 -1) =10^0.5, 選項(4)正確。
(ab)^√2 = (a‧a^√2)^√2 = (a^(1+√2))^√2 = a^(2+√2) = (10^(1 - √2/2))^(2+√2)
= 10^((1 - √2/2)(2 + √2)) = 10^(2 - √2 + √2 - 2/2) = 10 , 選項(5)錯誤。