### Video Transcript

Simplify π raised to the power two
over five multiplied by π to the power of a half to the power five all over π to
the power three over two multiplied by 32π to the power eight to the power one over
five, where π is a positive constant and π does not equal one.

Weβre given an exponential
expression where we have the positive base π, which is not equal to positive one,
to various powers or exponents within a fraction. And so we have various exponential
expressions within a rational expression. To simplify this, weβll use the
rules or laws of exponents. So letβs make a note of these
before we start.

First, we have the product rule,
where we saw our exponents. Second, we have the power rule,
where we multiply our exponents. Our third rule or law is the power
of products. Number four is the negative
exponent law. And number five is the law for
rational or fractional exponents. And in general, π and π are
nonzero integers, where in our case, as noted, π is not equal to one. So letβs see now how we can use
these laws to simplify the given expression.

Letβs consider first the
numerator. We have π raised to the power two
over five multiplied by π to the power of a half all to the power five. Using our second law, thatβs the
power rule, on the second term, π to the power of a half all to the power five, we
simply multiply our exponents. That is one over two multiplied by
five, which is five over two, so that our numerator is π to the power two over five
multiplied by π to the power five over two.

Now we can apply the first law,
that is, the product law. Where given a product of
exponential expressions with the same base, we add the powers. In our case, thatβs two over five
plus five over two, which is 29 over 10. Our numerator is then π raised to
the power 29 over 10.

So now letβs look at our
denominator. We have π raised to the power
three over two multiplied by 32π to the power eight all to the power one over
five. And for our second term here, we
can use rule number three, that is, the rule for a power of a product. And so our denominator is π to the
power three over two multiplied by 32 to the power one over five multiplied by π to
the power eight over five, where weβve used rule number two for π to the power
eight all to the power one over five. So weβve multiplied eight by one
over five.

The middle term here has a
numerical base. And we can simplify this using rule
number five for rational exponents. If we recall that two to the power
five is 32, then two is equal to 32 raised to the power one over five, which is the
fifth root of 32. So far then, our denominator is π
raised to the power three over two multiplied by two multiplied by π to the power
eight over five.

And so now if we bring the constant
two to the front, we see that weβre going to be able to use rule number one to
simplify our denominator further. And we do this by summing our
powers of π. So now we have two multiplied by π
raised to the power three over two plus eight over five. Three over two plus eight over five
is 31 over 10. And so our denominator is two
multiplied by π raised to the power 31 over 10.

So now making some space, our
expression is now simplified to π to the power 29 over 10 divided by two π to the
power 31 over 10. And now splitting this into a
product of fractions, we can use rule number four on one over π to the 31 over 10
so that one over π to the 31 over 10 is π to the negative 31 over 10. So now we have π to the 29 over 10
divided by two multiplied by π to the negative 31 over 10.

So, finally, we can again use our
product rule number one, where we sum our two powers of π. And 29 over 10 plus negative 31
over 10 is negative two over 10. And thatβs negative one over
five. And we have π raised to the power
negative one over five over two. So the given expression simplifies
to π raised to the power negative one over five over two.

We note that this could also be
written one over two π to the power one over five. Thatβs using rule number four. And this could also be written as
one over two times the fifth root of π using rule number five. However, we leave our answer in the
form π to the power negative one over five divided by two.