Given that five to the 𝑥 minus six
power times six to the six minus 𝑥 power is equal to 216 over 125, find the value
Here’s what we know. Before we do anything with the left
side of our equation, I want to break this fraction apart. I want to say 216 times one over
125. And then, I want to rewrite one
over 125 like this: 125 to the negative one power.
If you’re still not sure where I’m
going with this, hold on, it should make sense soon. Our statement now says five to the
𝑥 minus six times six to the six minus 𝑥 is equal to 216 times 125 to the negative
Now, I want to know can I write 216
as a power with the base of six and can I write 125 as a power with the base of
five? Six cubed is equal to 216 and five
cubed equals 125. So we can write five cubed to the
negative one power. Five cubed to the negative one
power can be simplified as five to the negative three. Six cubed times five to the
negative three is equal to five to the 𝑥 minus six times six to the six minus
What we can do now is set the
powers with the same bases equal to each other. The powers with a base five, we’ll
say 𝑥 minus six equals negative three. And the powers with the base six,
six minus 𝑥 is equal to three.
Remember that our goal is to find
the value of 𝑥. On the left, we add six to both
sides of the equation. Negative three plus six equals
three. 𝑥 equals three. On the right, subtract six from
both sides of the equation. Negative 𝑥 is equal to negative
three. We’re looking for positive 𝑥. We multiply both sides of the
equation by negative one. 𝑥 is equal to three.
To make this statement true, the
value of 𝑥 must be equal to three.