We're given the exponential equation, a raised to the power of b
equals b raised to the power of a. And a and b are different positive integers
We're asked, what is the value of a and what is the value b? Let's first list out the set
of positive integers.
A positive integer will be in the set, one,
two, three, four, five, six and so on.
So we want two different numbers from this set
that would satisfy the equation, a raised to the power of b
equals b raised to the power of a. Meaning the base on the left side of the equation
must be the same as the exponent on the right side. And the exponent on the left
side of the equation
must be the same as the base on the right.
So one way to try and solve this equation would be trial and error,
where we just select two different values from this set,
perform substitution and see if it works. Let's begin by doing this.
Let's let a equal one and b equal two.
So if this was our solution
that it must satisfy this equation, which would be one
to the second equals two to the first.
And this is not true because one squared, or one to the second is one.
and two to the first is just two and of course one doesn't equal two
and therefore, this is not our solution. Using trial and error again let's try a equals two
and b equals three. So if a equals two and b equals three,
we'd have the equation, two to the third
equals three to the second, which again is also not true.
Two to the third is equal to eight,
eight does not equal three square, which equals nine. Which means a equals two and
b equals three again is not our solution.
So thinking about this another way. If these two exponential expressions are equal to
each other,
they must be able to be written with the same base.
Which means we need to select a value of a and b that could be written
with the same base.
And because four can be written as two squared,
let's let a equal two and b equal four and see how this works.
We can write these with the common base of two if needed because four
is equal to two squared.
Performing substitution, a to the power of b would be two to the fourth
equals. B to the power of a would be
four to the second. I think this will actually work.
Two to the fourth means we have four factors of two, which is sixteen.
Four squared means we have two factors of four, which is also sixteen.
So this actually does satisfy our exponential equation
and therefore, the value of a is two and the value of b is four.
We could also show these are equal
by obtaining a common base. For example, if we start again with two to the fourth
equals four to the second. Again we could write four as two squared,
so we could write this as two to the fourth
equals two squared to the second,
again here's the four. When we have powers, the powers we multiply the exponents.
So here we have two to the fourth does equal two
raised to the power of two times two, which is four. Here's another way to show
that a equals two and b equals four is a solution.
Let's look at another example. Here we're given
a to the power of c equals b to the power of c
an a and b are different non-zero integers.
So if we look at the equation, a raised to the power of c equals b raised to the power of c,
Notice how here the two exponents are equal to each other,
but the bases we're told
are non-zero integers.
So how can we have different bases and the same exponent
and these two expressions be equal to each other? It's only going to happen
with one value of c,
and that's when c is equal to zero. One of the exponent rules is
a raised to the power of zero is equal to one
as long as a doesn't equal zero.
So if we select two different non-zero integers,
let's say we let a equal five and b equal two.
Performing substitution, with c equal to zero,
we would have five to the zero equals
two to the zero. And again both of these are equal to one, so we'd have
one equals one. It's also true if we have a negative integer
for example, if we try, let's say a equal negative three
and b equal nine and again c is zero. We would have a,
which is negative three in parenthesis, raised to the zero.
And we do need parenthesis, so that the exponent of zero is attached to
the base of negative three equals,
nine to the zero. And again both of these are equal to one.
So our answer here is c is zero.
Okay, I hope you found this helpful.