# uc3rmw_0301e01

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Hi there.
My name is Paul, and we're going to evaluate
exponential functions at different input
values.
This example has four parts.
Let's take a look at the first part.
Part A says, let f of x be equal
to 3 to the power x minus 2.
We want to evaluate f of x at x equal 4.
We substitute 4 in for x, so we have that f
of 4 is equal to 3 to the power 4 minus 2,
having replaced x with 4 in the exponent.
We simplify the exponent.
We have 4 minus 2 is 2, and so we're
evaluating 3 squared, which is 9.
And so our solution is
that f of 4 is equal to 9.
Now we move on to part B. We want to
evaluate the function g of x equal minus 2
times 10 to the x at x equals minus 2.
So as we did in part A,
we'll replace 6 with minus 2.
We have g of minus 2 is equal to
minus 2 times 10 to the minus 2 power.
The exponent is negative,
so we're going to take
the reciprocal of that factor to convert
it to a factor with a positive exponent.
So we have minus 2 times 1
over 10 to the second power.
Note that the minus 2 out front is not part
of the exponent factor in the expression.
As a positive exponent, the
factor is now easy to evaluate.
10 squared is 100.
So we have minus 2 times 1 over 100,
and that's equal to minus 2 over 100
or minus 0.02.
Therefore, g of minus 2
is equal to minus 0.02.
Let's take a look at part C. Here we want
to evaluate the function h of x equal 1/9
to the x power at x equal minus 3/2.
We replace x with minus 3/2,
and we have that h of minus 3/2
is equal to the fraction
1/9 to the power minus 3/2.
So we note that the base of this
exponent is the reciprocal of 9,
so we're going to express that as an
exponent 9 to the negative 1 power--
and all that taken to the minus 3/2 power.
At this point, we're going to
invoke the property of exponents,
where when we have something taken to
a power and then that whole thing taken
to another power, we go ahead
and multiply the exponents.
So we have 9 to the minus 1
times, in parentheses, minus 3/2.
Multiplying those two exponents
together, we have 9 to the 3/2 power.
At this point, we can go ahead and split
off the 1/2 power, take that first.
The 1/2 power becomes radical 9, and all
of that is being taken to the third power.
Radical 9, or square root
of 9, is simplified to 3,
and that's taken to the third power.
3 to the third power is 27.
So we have that h of
minus 3/2 is equal to 27.
So in the last part of this example, in part
D, we want to evaluate f of x equal to 4
to the x power at x equals 3.2.
So f of 3.2.
We'll go ahead and replace x with
3.2, so we have 4 to the 3.2 power.
In this case, there are
no exponent properties
that we can apply to manipulate
this expression to a form that
will allow us to evaluate this
expression without a calculator.
So we go ahead and plug
this into our calculator,
and we get that 4 to the 3.2 power
is approximately 84.44850629,
out to eight digits past the decimal point.
So that's it.
We just evaluated exponential functions
by using the laws of exponents
and also with a calculator.