### Video Transcript

Given that two to the power of π
multiplied by four to the power of π equals 2048 and two to the power of π
multiplied by four to the power of π is equal to 1024, determine the values of π
and π.

Well, the first thing we want to do
is rewrite it so that each of our terms is expressed as in powers of two. So, when we do that, weβve got two
to the power of π multiplied by two squared to the power of π equals 2048. And weβve got two to the power of
π multiplied by two squared to the power of π equals 1024.

Well, itβs here where we can apply
one of our exponent laws. And thatβs if we have π₯ to the
power of π all to the power of π, this is gonna be equal to π₯ to the power of
ππ. So, we can multiply the
exponents. So, in both of our examples, it
means that we can multiply two and π and two and π.

So, weβre gonna start with the
left-hand side first. So, weβve got two to the power of
π is equal to two to the power of two π, because we multiplied the two and the π
together, is equal to 2048. Well, therefore, weβve got two to
the power of π multiplied by two to the power of two π is equal to two to the
power of 11. And thatβs because 2048 is two to
the power of 11. But as we have the same base
throughout, what we can do is we can equate the exponents.

And to enable us to do that, weβre
gonna use the second exponent law. And this is that if we have π₯ to
the power of π multiplied by π₯ to the power of π, then we have π₯ to the power of
π plus π. So, we add the exponents, if we got
the same base and we multiply them together. So, in our case, weβre gonna get π
plus two π is equal to 11. And thatβs because we had two to
the power of π multiplied by two to power of two π is equal to two to the power of
11.

So, now weβre gonna do the same to
the right-hand side. So, first of all, we get two to the
power of π multiplied by two to the power of two π is equal to 1024. So therefore, weβre gonna have two
to the power of π multiplied by two to the power of two π is equal to two to the
power of 10. Thatβs because 1024 is two to the
power of 10. And we could see that because if we
divided 2048 by two, we get 1024.

And, once again, weβve got the same
base, so what we can do is we can equate the exponents. And we do that using the second
exponent law again. And therefore, when we do that, we
get π plus two π is equal to 10. So, now we have a pair of
simultaneous equations because weβve got two unknowns, thatβs π and π. And weβre gonna solve these to find
π and π.

And the first thing weβre gonna do
is weβre gonna subtract two π from both sides of equation two. And the reason weβre gonna do that
is weβre gonna solve this using the substitution method. And to solve it using the
substitution method, what we need to do is to change the subject so that either π
or π are the subject of the equation.

So, Iβve done that in this case and
Iβve made π the subject by subtracting two π from each side. So, Iβve got π equals 10 minus two
π. And weβre gonna call that equation
three. Iβve labelled each of the equations
because it makes it easier to see what weβre doing when we do it step-by-step.

So, now the next step is to
substitute equation three into equation one. So, when we do that, weβre gonna
get π plus two multiplied by then weβve got 10 minus two π. Thatβs because π was equal to 10
minus two π. And then this is equal to 11. And then, when we expand the
parentheses, we get π plus 20 minus four π equals 11. And then, when we tidy this up, we
get negative three π plus 20 is equal to 11. And then, what weβre gonna do is
solve this to find π.

So, weβre gonna add three π to
each side of the equation and subtract 11. And thatβs because we want positive
πs. And when we do that, we get nine is
equal to three π. And thatβs because if you add three
π onto both sides of the equation, we get three π on the right-hand side. And if we subtract 11, well, 11
away from 20 gives us nine. So, nine equals three π. And then, if we divide both sides
of the equation by three, we get π is equal to three. So, thatβs the first part
solved. Now what we need to do is to find
π.

Well, to find π, what we need to
do is substitute π equals three into equation three. So, weβre gonna get π is equal to
10 minus then youβve got two multiplied by three cause we had π is equal to 10
minus two π. So therefore, we get π is equal to
four. And thatβs because we had 10 minus
six which gives us four.

So, weβve solved the problem. And weβve found the values of π
and π. But what weβre gonna do now is just
quickly check it. Well, to check it, Iβm gonna
substitute our π- and π-values back in.

So, weβre gonna have two to the
power of three multiplied by four to the power of four. So, weβre gonna do the first
equation we had. Well, this is gonna give us eight
multiplied by 256. And thatβs cause two cubed is
eight. And four to the power of four is
256, which gives us an answer of 2048, which is the same as we were given in the
question. So therefore, we can say that,
given that two to the power of π multiplied by four to the power of π is equal to
2048 and two to the power of π multiplied by four to the power of π is equal to
1024, the values of π and π are three and four, respectively.