1. ## Arithmetic

If $\displaystyle 2^{2a}-2^{2b}=55.$.
What is the value of a and b?

2. Hello, Apprentice123!

If $\displaystyle 2^{2a}-2^{2b}=55$, what is the value of $\displaystyle a$ and $\displaystyle b$?
With one equation in two unknowns, there are infinite solutions.
. . I'll show you two of them . . .

We have: .$\displaystyle (2^a-2^b)(2^a + 2^b) \:=\:55$

Let: .$\displaystyle \begin{array}{cccc}2^a-2^b &=& 5 & [1] \\ 2^a + 2^b &=& 11 & [2] \end{array}$

Add [1] and [2]: .$\displaystyle 2\!\cdot\!2^a \:=\:16 \quad\Rightarrow\quad 2^a \:=\:8 \quad\Rightarrow\quad a \:=\:3$

Subtract [2] - [1]: .$\displaystyle 2\!\cdot\!2^b \:=\:6 \quad\Rightarrow\quad 2^b \:=\:3 \quad\Rightarrow\quad b \:=\:\log_2\!3$

. . Therefore: .$\displaystyle (a,b) \;=\;(3,\:\log_2\!3)$

Let: .$\displaystyle \begin{array}{cccc}2^a - 2^b &=& 1 & [3] \\ 2^a + 2^b &=& 55 & [4]\end{array}$

Add [1] and [2]: .$\displaystyle 2\!\cdot\!2^a \:=\:56\quad\Rightarrow\quad 2^a \:=\:28 \quad\Rightarrow\quad a \:=\:\log_2\!28$

Subtract [2] - [1]: .$\displaystyle 2\!\cdot\!2^b \:=\:54 \quad\Rightarrow\quad 2^b \:=\:27 \quad\Rightarrow\quad b \:=\:\log_2\!27$

. . Therefore: .$\displaystyle (a,b) \;=\;(\log_2\!28,\:\log_2\!27)$