Question

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.

Answer 1

C, E, A, B, D

Explanation:

I had this question and this was the correct answer

Answer 2

The order is "D, B, A, E, C".

Explanation:

The numbers are as follows:

`A=(2^(1/2))/(4^(1/6))\\\\B = \sqrt[12]{128}\\\\C=((1)/(8^(1/5)))^(2)\\\\D = \sqrt{(4^(-1))/(2^(-1)\cdot 8^(-1))}\\\\E = \sqrt[3]{2^(1/2)}\cdot 4^(-1/4)`

Simplify the value of A, B, C, D and E as follows:

`A=(2^(1/2))/(4^(1/6))=(2^(1/2))/(2^(2/6))=(2^(1/2))/(2^(1/3))=2^(1/2-1/3)=2^(1/6)=1.1225\\\\B = \sqrt[12]{128}=(128)^(1/12)=(2^(7))^(1/12)=2^(7/12)=1.4983\\\\C=((1)/(8^(1/5)))^(2)=((1)/((2^(3))^(1/5)))^(2)=((1)/(2^(3/5)))^(2)=(1)/(2^(3/5*2))=(1)/(2^(6/5))=2^(-6/5)=0.4353\\\\D = \sqrt{(4^(-1))/(2^(-1)\cdot 8^(-1))}=\sqrt{(2^(-2))/(2^(-1)\cdot 2^(-3))}=\sqrt{2^(-2+1+3)}=2\\\\E = \sqrt[3]{2^(1/2)}\cdot 4^(-1/4)= (2^(1/2))^(1/3)\cdot 2^(-2/4)=2^(-1/3)=0.7937`

Arrange the following numbers in increasing order as follows:

D > B > A > E > C

Thus, the order is "D, B, A, E, C".