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To solve the equation
, we can start by simplifying the left side.
First, we can use the property of exponents that states
. This gives us:
Now, we can rewrite the left side of the equation:
Now the equation becomes:
Next, we express 1728 as a power of 2. We can divide 1728 by 2 repeatedly to find its prime factorization:
Now, we need to equate the powers of 2. We have:
Since the right side has a factor of
which cannot be expressed as a power of 2, it shows that
(which is not purely a power of 2) does not yield any integer solutions for
.
Upon re-evaluating, if we want a numerical approach, we can isolate
instead, or reconsider its components.
As such, with
, our target then will be specific to the base
. Thus, we derive
.
Setting the components equal gives:
Thus, our solution is: