### Video Transcript

Find the solution set in the set of real numbers for two 𝑥 plus one to the power of four is equal to negative 81.

Let’s consider what happens as we begin to solve this equation. Firstly, the inverse operation to finding the fourth power is taking the fourth root. When we take the fourth root of both sides of this equation, we have two 𝑥 minus one is equal to the fourth root of negative 81. The next step would be to see if we could find the fourth root of negative 81.

At this point, we might realize that we have a problem because we can’t take the fourth root of negative 81. We can define this problem more formally by saying that for 𝑦 to the power of 𝑛 is equal to 𝑥, with 𝑥 and 𝑦 in the real numbers, if 𝑥 is less than zero and 𝑛 is even, there are no real solutions. Here, we are trying to find a value of 𝑦 such that 𝑦 to the power of four is equal to negative 81. The value of 𝑥, which is negative 81, is less than zero. And, importantly, 𝑛 is even. That’s the value of four. And so that’s why there are no real solutions.

As an aside, you might remember that the fourth root of positive 81 is three. So why is the fourth root of negative 81 not negative three? Well, let’s remember that if we wanted to work out negative three to the power of four, that’s equal to negative three times negative three times negative three times negative three. When we work this out in smaller parts, negative three times negative three will give us a positive value of nine. Then, we have another negative three times negative three, which also gives us positive nine. And multiplying nine times nine gives us a positive value of 81. And so there can’t be a real value for the fourth root of negative 81.

Because the question has asked us to find a solution in the set of real numbers and there are none, then the answer has to be the empty set or the null set, because there are no real solutions to this problem.