Evaluate 27 to the power of negative one-third.
In this question, we can see that we have a negative exponent of negative a third. So let’s start by changing this negative exponent into a positive one. To do this, we’re going to use the exponent rule that if we have a negative exponent, for example, 𝑎 to the power of negative 𝑛, then we can write this with a positive exponent as one over 𝑎 to the power of 𝑛. So taking our 27 to the power of negative one-third, we can write this as one over 27 to the power of one-third.
Considering our value 27 to the power of one-third, what does that actually mean? Well, it’s the same as the cube root of 27. So for the cube root of 27, we’re trying to find a value 𝑥 that will give us 𝑥 times 𝑥 times 𝑥 equals 27. Since we know that three times three times three will give us 27, then the cube root of 27 is three. So our complete value will be one over three. And therefore 27 to the power of negative one-third is equal to one-third.