Question Video: Finding the Solution Set of an Exponential Equation Mathematics

Video Transcript

Determine the solution set of 𝑥 to the power of 𝑥 squared minus 64 equals six to the power of 𝑥 squared minus 64.

Now let’s begin by noticing that the exponents on each side of our equation are in fact equal. And so, for both sides of our equation to be equal, one value of 𝑥 would be when the bases, the big numbers, are equal. In other words, if we have 𝑥 is equal to six, both sides of our equation are identical, so that is one solution. But are there any other options? Well, another way to ensure that two sides of our equation are equal is to have a power of zero since anything to the power of zero is one. And so what we could do is say that the exponent 𝑥 squared minus 64 is equal to zero.

Let’s solve for 𝑥 by adding 64 to both sides to give us 𝑥 squared equals 64. And then, finally, we take the square root of both sides, remembering, of course, to take both the positive and negative square root of 64. But since the square root of 64 is eight, we can say that the solutions to this equation are 𝑥 equals negative or positive eight. And so, so far, we have three possible values of 𝑥. But in fact, there is one more. And that solution is 𝑥 equals negative six. So, why does 𝑥 equals negative six work?

Well, imagine 𝑥 is equal to negative six. When it is, the exponent becomes negative six squared minus 64, which is negative 28. And we know that if 𝑎 is an even number, the negative 𝑥 to the power of 𝑎 is the same as 𝑥 to the power of 𝑎. And so, the left-hand side is negative six to the power of negative 28. But because negative 28 is even, this is the same as six to the power of negative 28, which is what we get on the right-hand side. And so, 𝑥 equals negative six also has to be a solution. Now, of course, we only chose negative six because we wanted to match the bases. We wouldn’t go ahead and choose any value of 𝑥 that would make the exponents equal. It only works for negative six.

And so we can say the solution set to our equation is the set containing the elements six, negative six, eight, and negative eight.