+73 Arrange the following numbers in increasing order:
\(\begin{align*} A &= \frac{2^{1/2}}{4^{1/6}}\\ B &= \sqrt[12]{128}\vphantom{dfrac{2}{2}}\\ C &= \left( \frac{1}{8^{1/5}} \right)^2\\ D &= \sqrt{\frac{4^{-1}}{2^{-1} \cdot 8^{-1}}}\\ E &= \sqrt[3]{2^{1/2} \cdot 4^{-1/4}}.\vphantom{dfrac{2}{2}} \end{align*}\)
Enter the letters, separated by commas. For example, if you think that \(D < A < E < C < B\), then enter "D,A,E,C,B", without the quotation marks.
A 2^(1/2) / 4^(1/6) = 2^(1/2) / ( 2^2)^(1/6) = 2^(1/2) / 2^(1/3) = 2^(1/2 - 1/3) = 2^(1/6)
B (128)^(1/28) = [ 2^(7)]^(1/28) = 2^(1/4)
C (1 / 8^(1/5))^2 = [ (8)^(-1/5)]^2 = (2^3)^(-1/5) = 2^(-3/5)
D Note 4^(-1) = 1/4 2^(-1) = 1/2 8^(-1) = 1/8
So what we have inside the radical is just this :
1/4 1/4
_________ = _______ = (1/4) * (16/1) = 4
1/2 * 1/8 1/16
And the sqrt of this = 2
E Inside the radical we have
2^(1/2) * 4^(-1/4) =
2^(1/2) * (2^2)^(-1/4) =
2^(1/2) * 2^(-1/2) =
2^(1/2 - 1/2) =
2^0 =
1
And the cube root of this is still just 1
So in order we have
2^(-3/5) , 2^(1/6) , 2^(1/4) , 1 , 2
C , A , B , E , D
+9466 The above answer was copy and pasted from here: https://web2.0calc.com/questions/a-good-math-question
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