Video Transcript
Find the value of 𝑥 for which 81
to the power of 𝑥 plus five is equal to one-third to the power of 𝑥.
We’ve been given an equation
involving two terms with variable exponents. We have 81 to the power of 𝑥 plus
five and one-third to the power of 𝑥. The key to answering this question
is to spot that both 81 and one-third can be written as powers of three. In fact, 81 is three to the fourth
power. And since a negative exponent gives
us that reciprocal, one-third is three to the power of negative one. This means we can rewrite 81 to the
power of 𝑥 plus five as three to the fourth power to the power of 𝑥 plus five.
And this is really useful because
we know that when we have an expression raised to a power and then we raise that to
another power, we simply multiply those powers or those exponents. So 𝑥 to the power of 𝑎 to the
power of 𝑏 is 𝑥 to the power of 𝑎 times 𝑏. So three to the power of four to
the power of 𝑥 plus five is the same as three to the power of four times 𝑥 plus
five.
Let’s repeat this process with
one-third to the power of 𝑥. It’s three to the power of negative
one times 𝑥. And then we’re going to multiply
those exponents. Well, negative one times 𝑥 is
negative 𝑥. So one-third to the power of 𝑥 is
the same as three to the power of negative 𝑥.
And so we can rewrite our original
equation as three to the power of four times 𝑥 plus five equals three to the power
of negative 𝑥. And this is great because we now
have two expressions with equal bases. So for these two expressions to be
equal, their powers themselves must also be equal. That is, four times 𝑥 plus five
must be equal to negative 𝑥. Let’s distribute the parentheses on
the left-hand side, and we’ll then be able to solve for 𝑥.
When we do, we get four 𝑥 plus 20
equals negative 𝑥. Let’s subtract four 𝑥 from both
sides. So 20 is equal to negative five
𝑥. To solve for 𝑥, we now need to
divide through by negative five. So 𝑥 is 20 divided by negative
five. And that of course is equal to
negative four. The value of 𝑥 then for which 81
to the power of 𝑥 plus five is equal to one-third to the power of 𝑥 is negative
four.
