### Video Transcript

Given that three to the power of ๐ฅ
equals 27 and three to the power of ๐ฅ plus ๐ฆ equals 81, find the value of ๐ฅ and
the value of ๐ฆ.

Well, thereโre a couple of ways we
go about solving it. One way is to use an exponent
rule. And that rule is a multiplication
rule, which tells us that if we have ๐ฅ to the power of ๐ multiplied by ๐ฅ to the
power of ๐, itโs equal to ๐ฅ to the power of ๐ plus ๐. So therefore, using this, what we
can do is rewrite three to the power of ๐ฅ plus ๐ฆ equals 81 as three to the power
of ๐ฅ multiplied by three to the power ๐ฆ is equal to 81.

And now, what we can do is we can
substitute in three to the power of ๐ฅ equals 27 into our equation. And if we do that, what we get is
27 multiplied by three to the power of ๐ฆ equals 81. So, now what we can do is divide
both sides of our equation by 27 to find out what three to the power of ๐ฆ is. When we do that, we get three to
the power of ๐ฆ. And thatโs because if we divide the
left-hand side by 27, weโre left with three to the power of ๐ฆ. And we get itโs equal to three, and
thatโs because 81 divided by 27 is three.

Well, if three to the power of ๐ฆ
is equal to three, therefore ๐ฆ must be equal to one. And thatโs because three to the
power of one is three. Also, we can think about it as if
we equate our exponents at this point, weโd have ๐ฆ is equal to one. So, thatโs our value of ๐ฆ
found. And now, we want to find our value
of ๐ฅ. Well, we can say that three
multiplied by three multiplied by three is equal to 27. So therefore, three cubed is equal
to 27. So therefore, we can say that ๐ฅ is
equal to three and ๐ฆ is equal to one.

What we can do is quickly check
this. And we can do that by substituting
in ๐ฅ equals three and ๐ฆ equals one into three to the power of ๐ฅ plus ๐ฆ equals
81. So, we do that, weโre gonna get
three to the power of three plus one equals 81, which gives us three to the power of
four is equal to 81. Or, three multiplied by three is
nine multiplied by another three is 27 then multiplied by a fourth three is 81. So yes, this is correct. We can definitely say that ๐ฅ
equals three and ๐ฆ equals one.