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# Manipulating Exponents

-1
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Write $$\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{3}{m}}\right)^3$$as a power of 2.

Jan 14, 2022

#1
+117441
+2

$$\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{3}{m}}\right)^3\\ \frac{4}{27}\left((2^{\frac{3+m}{3}})^3 + 3 (2^{\frac{3+m}{3}})^2 (2^{\frac{3}{m}}) +3(2^{\frac{3+m}{3}}) (2^{\frac{3}{m}})^2 + (2^{\frac{3}{m}})^3 \right)\\ \frac{4}{27}\left((2^{(3+m)} + 3 (2^{(1+\frac{m}{3})})^2 (2^{\frac{3}{m}}) +3(2^{1+\frac{m}{3}}) (2^{\frac{6}{m}}) + (2^{\frac{9}{m}}) \right)\\ \frac{4}{27}\left((2^{(3+m)} + 3 (2^{(2+\frac{2m}{3})}) (2^{\frac{3}{m}}) +3(2^{1+\frac{m}{3}}) (2^{\frac{6}{m}}) + (2^{\frac{9}{m}}) \right)\\ \frac{2^2}{3^3}\left((2^32^m + 3 (2^2 2^{(\frac{2m}{3})}) (2^{\frac{3}{m}}) +3(2* 2^{(\frac{m}{3})}) (2^{\frac{6}{m}}) + (2^{\frac{9}{m}}) \right)\\$$

I can keep going but this is not going to fall out nicely.

LaTex:

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

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

\frac{4}{27}\left((2^{(3+m)}  +  3 (2^{(1+\frac{m}{3})})^2  (2^{\frac{3}{m}})  +3(2^{1+\frac{m}{3}})  (2^{\frac{6}{m}})  +   (2^{\frac{9}{m}}) \right)\\

\frac{4}{27}\left((2^{(3+m)}  +  3 (2^{(2+\frac{2m}{3})})  (2^{\frac{3}{m}})  +3(2^{1+\frac{m}{3}})  (2^{\frac{6}{m}})  +   (2^{\frac{9}{m}}) \right)\\

\frac{2^2}{3^3}\left((2^32^m  +  3 (2^2 2^{(\frac{2m}{3})})  (2^{\frac{3}{m}})  +3(2* 2^{(\frac{m}{3})})  (2^{\frac{6}{m}})  +   (2^{\frac{9}{m}}) \right)\\

Jan 15, 2022
#2
+2363
+2

Here’s a neat and orderly way to transcribe this as a power of (2):

$$\hspace {15em} \huge2 \large^ {\log_2\left(\frac{4}{27}\left(2^{\frac{3+m}{3}}+2^{\frac{3}{m}}\right)^3\right)}\\$$

GA

--. .-

Jan 15, 2022
#3
+117441
+1

LOL you got me there Ginger !

Melody  Jan 15, 2022
#4
+176
0

Guys im confused.

From trials and error I found that: $$2^{m+2}$$ works for some reason.

Jan 16, 2022
#5
+117441
0

Hi Goo

Show us why you think   2^(m+2) is, or might be,  correct.

Are you sure you wrote the question down right?

Melody  Jan 16, 2022