# Simple Algebraic Manipulation

#### 1337h4x

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
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 $$\displaystyle k = 2^m$$...

#### 1337h4x

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
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.

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

$$\displaystyle a^m \cdot a^n=a^{m+n}$$