Simple Algebraic Manipulation

Apr 2009
96
0
Can somebody explain the rules of algebra so that the following expression makes sense:

\(\displaystyle 2^{2 \cdot 2^m}+1\) \(\displaystyle =\left(2^{2^m}\right)^2+1\)

I thought that \(\displaystyle \left(2^{2^m}\right)^2+1\) \(\displaystyle =4^{ \cdot 4^(m^2)}+1\) but apparently thats wrong.
 

Isomorphism

MHF Hall of Honor
Dec 2007
1,465
674
IISc, Bangalore
Can somebody explain the rules of algebra so that the following expression makes sense:

\(\displaystyle 2^{2 \cdot 2^m}+1\) \(\displaystyle =\left(2^{2^m}\right)^2+1\)

I thought that \(\displaystyle \left(2^{2^m}\right)^2+1\) \(\displaystyle =4^{ \cdot 4^(m^2)}+1\) but apparently thats wrong.
The rule is \(\displaystyle (a^l)^k= a^{lk}\).

Now choose \(\displaystyle a = 2, l = 2\) and \(\displaystyle k = 2^m\)...
 
Apr 2009
96
0
The rule is \(\displaystyle (a^l)^k= a^{lk}\).

Now choose \(\displaystyle a = 2, l = 2\) and \(\displaystyle k = 2^m\)...
I'm not sure how we get from my form of \(\displaystyle \left(a^{b^m}\right)^k\) to \(\displaystyle (a^l)^k\)
 

masters

MHF Helper
Jan 2008
2,550
1,187
Big Stone Gap, Virginia
Can somebody explain the rules of algebra so that the following expression makes sense:



I thought that but apparently thats wrong.
Hi 1337h4x,

This is what it looks like to me.

\(\displaystyle 2^{2 \cdot 2^m}+1=\left(2^{2^m}\right)^2\)

\(\displaystyle 2^{2^1 \cdot 2^m}+1=2^{2^m \cdot 2}\)

\(\displaystyle 2^{2^{m+1}}+1=2^{2^{m+1}}+1\)

The properties you use is the power of a power rule and the product of powers rule.

\(\displaystyle (a^m)^n=a^{mn}\)

\(\displaystyle a^m \cdot a^n=a^{m+n}\)