Please find the original thread, and answer here: https://web2.0calc.com/questions/manipulating-exponents
Melody the question I wrote was wrong. It was actually:
Write \(\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{m}{3}}\right)^3\)as a power of 2.
Thanks Mathy.
Now it comes out perfectly well.
If you still need help with it please put this correct version, along with the explantatory note, on the original thread.
I can help you over there is you still want me too,
\(\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{m}{3}}\right)^3\\ =\frac{4}{27}\left( \left[ 2^{\frac{3+m}{3}} \right] + \left[ 2^{\frac{m}{3}} \right] \right)^3\\ =\frac{4}{27}\left(\left[ 2^{\frac{3+m}{3}} \right]^3 + 3*\left[ 2^{\frac{3+m}{3}} \right]^2 \left[ 2^{\frac{m}{3}} \right] + 3*\left[ 2^{\frac{3+m}{3}} \right] \left[ 2^{\frac{m}{3}} \right] ^2 + \left[ 2^{\frac{m}{3}} \right] ^3 \right)\\ =\frac{4}{27}\left(\left[ 2^{3+m} \right] + 3*\left[ 2^{\frac{6+2m}{3}} \right] \left[ 2^{\frac{m}{3}} \right] + 3*\left[ 2^{\frac{3+m}{3}} \right] \left[ 2^{\frac{2m}{3}} \right] + \left[ 2^{m} \right] \right)\\ =\frac{4}{27}\left(\left[ 2^{3+m} \right] + 3*\left[ 2^{\frac{6+3m}{3}} \right] + 3*\left[ 2^{\frac{3+3m}{3}} \right] + \left[ 2^{m} \right] \right)\\ =\frac{4}{27}\left(\left[ 2^{3+m} \right] + 3*\left[ 2^{2+m} \right] + 3*\left[ 2^{1+m} \right] + \left[ 2^{m} \right] \right)\\ =\frac{4*2^m}{27}\left(\left[ 2^{3} \right] + 3*\left[ 2^{2} \right] + 3*\left[ 2 \right] + \left[ 1 \right] \right)\\ =\frac{4*2^m}{27}\left(8+ 12 + 6 + 1\right)\\ =\frac{4*2^m}{27}\left(27\right)\\ =2^{(m+2)} \)
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)}