Melody

\(\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{m}{3}}\right)^3\\ =\frac{4}{27}\left( \left[ 2^{\frac{3+m}{3}} \right] + \left[ 2^{\frac{m}{3}} \right] \right)^3\\ =\frac{4}{27}\left(\left[ 2^{\frac{3+m}{3}} \right]^3 + 3*\left[ 2^{\frac{3+m}{3}} \right]^2 \left[ 2^{\frac{m}{3}} \right] + 3*\left[ 2^{\frac{3+m}{3}} \right] \left[ 2^{\frac{m}{3}} \right] ^2 + \left[ 2^{\frac{m}{3}} \right] ^3 \right)\\ =\frac{4}{27}\left(\left[ 2^{3+m} \right] + 3*\left[ 2^{\frac{6+2m}{3}} \right] \left[ 2^{\frac{m}{3}} \right] + 3*\left[ 2^{\frac{3+m}{3}} \right] \left[ 2^{\frac{2m}{3}} \right] + \left[ 2^{m} \right] \right)\\ =\frac{4}{27}\left(\left[ 2^{3+m} \right] + 3*\left[ 2^{\frac{6+3m}{3}} \right] + 3*\left[ 2^{\frac{3+3m}{3}} \right] + \left[ 2^{m} \right] \right)\\ =\frac{4}{27}\left(\left[ 2^{3+m} \right] + 3*\left[ 2^{2+m} \right] + 3*\left[ 2^{1+m} \right] + \left[ 2^{m} \right] \right)\\ =\frac{4*2^m}{27}\left(\left[ 2^{3} \right] + 3*\left[ 2^{2} \right] + 3*\left[ 2 \right] + \left[ 1 \right] \right)\\ =\frac{4*2^m}{27}\left(8+ 12 + 6 + 1\right)\\ =\frac{4*2^m}{27}\left(27\right)\\ =2^{(m+2)} \)

LaTex

\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{m}{3}}\right)^3\\

=\frac{4}{27}\left(     \left[  2^{\frac{3+m}{3}} \right]    +    \left[ 2^{\frac{m}{3}} \right]    \right)^3\\
=\frac{4}{27}\left(\left[  2^{\frac{3+m}{3}} \right]^3 
+ 3*\left[  2^{\frac{3+m}{3}} \right]^2   \left[ 2^{\frac{m}{3}} \right]     
+ 3*\left[  2^{\frac{3+m}{3}} \right]   \left[ 2^{\frac{m}{3}} \right]  ^2  +  \left[ 2^{\frac{m}{3}} \right] ^3 \right)\\

=\frac{4}{27}\left(\left[  2^{3+m} \right] 
+ 3*\left[  2^{\frac{6+2m}{3}} \right]   \left[ 2^{\frac{m}{3}} \right]     
+ 3*\left[  2^{\frac{3+m}{3}} \right]   \left[ 2^{\frac{2m}{3}} \right]   +  \left[ 2^{m} \right]  \right)\\

=\frac{4}{27}\left(\left[  2^{3+m} \right] 
+ 3*\left[  2^{\frac{6+3m}{3}} \right]      
+ 3*\left[  2^{\frac{3+3m}{3}} \right]     +  \left[ 2^{m} \right]  \right)\\

=\frac{4}{27}\left(\left[  2^{3+m} \right] 
+ 3*\left[  2^{2+m} \right]      
+ 3*\left[  2^{1+m} \right]     +  \left[ 2^{m} \right]  \right)\\

=\frac{4*2^m}{27}\left(\left[  2^{3} \right] 
+ 3*\left[  2^{2} \right]      
+ 3*\left[  2 \right]     +  \left[ 1 \right]  \right)\\

=\frac{4*2^m}{27}\left(8+  12 + 6    + 1\right)\\
=\frac{4*2^m}{27}\left(27\right)\\
=2^{(m+2)}

What has -pi/2 got to do with anything.

Please put the question altogether properly.

Attn: Ora2004

You ask questions but you have never once responded to an answer.

I would not encourage people to answer you.  You are totally ungratful and are quite likely cheating.

If you were trying to learn you would be interacting with the answerers.

What is the question?

ok  :)

Where was that?

If you want me to see it you need to incluse a link.

(I should hae left a link too.)

Did you see my edit on our last post.  The answer has to be positive.  Take  a look.

Well .. are you going to show us how you did it?   OR guide us ??

That is pretty Good going!

Congratulations!!

I know what formula you are after but I can never remember  them of the top of my head so...

Let

\(\theta = acos(\frac{-1}{9})\\ \quad note: cos \theta = \frac{-1}{9}\\ find \;\;sin\frac{\theta }{2}\)

--------------------------------------------

\(cos(\frac{\theta}{2}+\frac{\theta}{2})=cos^2\frac{\theta}{2}-sin^2\frac{\theta}{2}\\ cos(\frac{\theta}{2}+\frac{\theta}{2})=1-sin^2\frac{\theta}{2}-sin^2\frac{\theta}{2}\\ cos\theta=1-2sin^2\frac{\theta}{2}\\ \frac{1-cos\theta}{2}=sin^2\frac{\theta}{2}\\ sin\frac{\theta}{2}=\pm \sqrt{\frac{1-cos\theta}{2}}\\ sin\frac{\theta}{2}=\pm \sqrt{\frac{1-\frac{-1}{9}}{2}}\\ sin\frac{\theta}{2}=\pm \sqrt{\frac{\frac{10}{9}}{2}}\\ sin\frac{\theta}{2}=\pm \sqrt{\frac{10}{18}}\\ sin\frac{\theta}{2}=\pm\sqrt{\frac{5}{9}}\\ sin\frac{\theta}{2}=\pm \frac{\sqrt5}{3}\\ sin\left[\frac{acos(\frac{-1}{9})}{2}\right]=\pm\frac{\sqrt5}{3}\\\)

Theta has to be in the second quadrant but theta/2 can be in the first or second quadrent .

Either way.   sin(theta) is positive!

\(\boxed{sin\left[\frac{acos(\frac{-1}{9})}{2}\right]=\frac{\sqrt5}{3}}\)

LaTex:

cos(\frac{\theta}{2}+\frac{\theta}{2})=cos^2\frac{\theta}{2}-sin^2\frac{\theta}{2}\\
cos(\frac{\theta}{2}+\frac{\theta}{2})=1-sin^2\frac{\theta}{2}-sin^2\frac{\theta}{2}\\
cos\theta=1-2sin^2\frac{\theta}{2}\\
\frac{1-cos\theta}{2}=sin^2\frac{\theta}{2}\\
sin\frac{\theta}{2}=\pm \sqrt{\frac{1-cos\theta}{2}}\\
sin\frac{\theta}{2}=\pm \sqrt{\frac{1-\frac{-1}{9}}{2}}\\
sin\frac{\theta}{2}=\pm \sqrt{\frac{\frac{10}{9}}{2}}\\
sin\frac{\theta}{2}=\pm \sqrt{\frac{10}{18}}\\
sin\frac{\theta}{2}=\pm\sqrt{\frac{5}{9}}\\
sin\frac{\theta}{2}=\pm \frac{\sqrt5}{3}\\
sin\left[\frac{acos(\frac{-1}{9})}{2}\right]=\frac{\sqrt5}{3}\\