Video Transcript
Find the solution set in the real numbers for 𝑥 squared minus five to the power of four is equal to 256.
The first step in finding the solution for 𝑥 is to take the inverse operation to the fourth power, which is finding the fourth root of both sides of the equation. This gives us 𝑥 squared minus five is equal to the fourth root of 256. At this point, we should pause when we’re working out 𝑛th roots and consider if there might be more than one solution to an 𝑛th root. If we have 𝑦 to the power of 𝑛 is equal to 𝑥 for real values of 𝑥 and 𝑦, with 𝑥 greater than zero and 𝑛 is even, then the solutions are 𝑦 is equal to plus or minus the 𝑛th root of 𝑥.
At this point, we’re trying to find the fourth root of 256, which means that we’re trying to find a value of 𝑦 such that 𝑦 to the power of four is equal to 256. Because we have a value of 256 — that’s the value of 𝑥 in the standard equation — greater than zero and the index 𝑛 is even, it’s four, then we have two solutions: 𝑦 equals plus or minus the fourth root of 256. And so let’s add the plus/minus sign into the workings on the right-hand side.
We can then work out that the fourth root of 256 is four. So we have 𝑥 squared minus five is equal to plus or minus four. Since the plus/minus sign indicates that there are two possible values, this means that we have two equations to solve: 𝑥 squared minus five equals positive four and 𝑥 squared minus five equals negative four. We can take each equation in turn and solve it.
For the equation on the left, we can add five to both sides, which gives us that 𝑥 squared is equal to nine. For the equation on the right-hand side, we also add five to both sides. But this time, negative four plus five will give us one. So now what we have, 𝑥 squared equals nine or 𝑥 squared equals one.
Before we do any further simplification, let’s consider if this highlighted statement will apply once more. Let’s take 𝑥 squared is equal to nine. The value of nine is positive, and the exponent two is even. And so when we take the square root of both sides, we won’t just have 𝑥 is equal to the square root of nine. But we’ll have 𝑥 is equal to plus or minus the square root of nine.
If we carry on solving this equation, we have that 𝑥 is equal to plus or minus three. Then, let’s take the equation 𝑥 squared is equal to one. Once again, the value of one is positive, and the index of two is even. Taking the square root of both sides, we should write that 𝑥 is equal to plus or minus the square root of one. Evaluating this, we have that 𝑥 is equal to plus or minus one. We can then give these four solutions for 𝑥 in set notation as the set containing one, three, negative one, and negative three, any one of these four values which when substituted for 𝑥 in the expression 𝑥 squared minus five to the power of four would give an answer of 256.
