Video Transcript
Evaluate 16 to the negative one-half power times eight to the negative two-thirds power times one over 16 to the negative one-half power.
When we’re multiplying values that involve fractions, we multiply the numerators together and the denominators together. 16 to the negative one-half power times eight to the negative two-thirds times one over 16 to the negative one-half power. Now, we have 16 to the negative one-half power over 16 to the negative one-half power. It’s true that 16 to the negative one-half power equals one over 16 to the one-half power and that one over 16 to the negative one-half power is equal to 16 to the positive one-half power. However, both ways show us that these values are equal to one. And that means our expression is equal to eight to the negative two-thirds power.
There are a few ways we could solve this. But before we do that, I want to think about this property, that 𝑎 to the 𝑥 power to the 𝑦 power is equal to 𝑎 to the 𝑥 times 𝑦 power. This is called a power of a power. It also means that there are a few ways we can rewrite eight to the negative two-thirds power. We could say that this is equal to eight to the two-thirds power to the negative one power because two-thirds times negative one is negative two-thirds. But we can separate this even further. We can say eight to the one-third power squared to the negative one power. This is because one-third times two times negative one equals negative two-thirds.
When we break the values down like this, it becomes a little bit simpler to operate. And so, by following the order of operations, we’ll first solve eight to the one-third power. Eight to the one-third power is asking what value cubed equals eight. We know that two times two times two equals eight. And that means eight to the one-third power equals two. And so we plug in two in place of eight to the one-third power. And we now have two squared to the negative one power. The next parentheses would say two squared equals four. And we’re now looking for four to the negative one power, which is one over four because 𝑎 to the negative 𝑥 power is equal to one over 𝑎 to the 𝑥 power. This expression is equal to one-fourth.
