+144 Without using a calculator, order the following numbers from least to greatest:
\(A= \frac{2^{1/2}}{4^{1/6}}\\ B= \sqrt[12]{128}\vphantom{dfrac{2}{2}}\\ C= \left( \frac{1}{8^{1/5}} \right)^2\\ \)
\(D= \sqrt{\frac{4^{-1}}{2^{-1} \cdot 8^{-1}}}\\\)
\(E= \sqrt[3]{2^{1/2} *4^{-1/4}}\)
Give your answer as a list of capital letters separated by commas. For example, if you think that , then you would answer E,B,C,D,A.
+128474 Put everything in terms of powers on 2
A = 2^(1/2) / 4^(1/6) = 2^(1/2) / (2^2)^(1/6) = 2^(1/2) / 2^(2/6) = 2^(3/6) / 2^(2/6) =
2^(1/6) = 2^(2/12)
B = (128)^(1/12) = (2^7)^(1/12) = 2^(7/12)
C = [ 8^(-1/5)] ^2 = [ (2^3)^(-1/5) ]^2 = [ 2^(-3/5) ]^2 = 2^(-6/5)
D = √ [ (2)^1 * 8^(1) / 4^(1) ] = √ [16 / 4 ] = √ 4 = 2 = 2^(1)
E = [ 2^(1/2) * (2^2)^(-1/4) ] ^(1/3) = [ 2^(1/2) * 2^(-2/4) ]^(1/3) =
[ 2^(1/2) * 2^(-1/2) ]^(1/3) = [ 2^0]^(1/3) = 2^0
So..in order, we have
2^(-6/5) < 2^(0) < 2^(2/12) < 2^(7/12) < 2^(1)
So
C E A B D
