How does one solve ${36^{x - 9}} = {6^{2x}}$ ?

Hint:For solving this particular problem we must create the expression equivalent that they have equal bases, after creating equivalent expressions by base we can say that the exponents of the expressions are also equal. Here we have the property when ${a^x} = {a^y} \Rightarrow x = y$ , and then simplify it further.

Formula Used:
we use the following formula ,
${a^x} = {a^y} \Rightarrow x = y$
Here we see that for using this above exponential formula , bases of expressions must be equal.

Complete step by step answer:
We have to solve ${36^{x - 9}} = {6^{2x}}$,
Firstly, we use the following formula,
${a^x} = {a^y} \\
\Rightarrow x = y$
Here we see that for using this above exponential formula , bases of expressions must be equal.Therefore, we have to make the bases equal. Now we know that we can convert $36$ into base $6$ form as,
$36 = {6^2}$
Now substitute this in our given expression,
${36^{x - 9}} = {6^{2x}}$
$ \Rightarrow {6^{2(x - 9)}} = {6^{2x}}$
Now applying the exponential formula , we will get,
$\Rightarrow 2(x - 9) = 2x \\
\therefore x - 9 = x $
And this is not possible.Therefore, we have no possible solution for this particular problem.

Note:Sign gets changed when we transpose the term which is present at one side to the other side of the equation. Here the term transpose means moving something from one side of the equation to the other side of the equation.When we do arithmetic operations on the balanced equation we must follow certain rules such as when we add the same number to both sides of the equation , the equation remains the same. When we subtract the same number from both sides of the equation, the equation remains the same.