Video Transcript
Find the solution set in the real numbers for two 𝑥 minus one to the power of six is equal to 64.
So let’s see how we would go about solving this equation to find the values of 𝑥. On the left-hand side, we can see that there is a power of six. And so the first operation that we must perform is the inverse operation. The inverse operation to the sixth power is taking the sixth root. When we take the sixth root of both sides of this equation, we have two 𝑥 minus one is equal to plus or minus the sixth root of 64.
Notice that we have a plus minus sign in front of the sixth root. This is because for any 𝑦 to the power of 𝑛 is equal to 𝑥 and 𝑥 and 𝑦 are real numbers, if 𝑥 is greater than zero and 𝑛 is even, then the solutions are 𝑦 equals plus or minus the 𝑛th root of 𝑥. Remember that if we’re trying to find something like the sixth root of 64, we’re really trying to find some value 𝑦 such that 𝑦 to the power of six is equal to 64. The value of 64 is greater than zero. And notice that the value of 𝑛, the exponent of six, is even. And so the solution must be 𝑦 equals plus or minus the sixth root of 64.
On the right-hand side then, we can work out that the sixth root of 64 is two. And we must include the plus or minus sign. We now have two equations: two 𝑥 minus one equals positive two or two 𝑥 minus one equals negative two. We solve both of these equations to work out the different values that 𝑥 can take. For the left equation, we add one to both sides of this equation, giving us two 𝑥 equals three. Dividing both sides by two, we have 𝑥 is equal to three over two. Then, to solve the second equation, we would also begin by adding one to both sides, giving us two 𝑥 is equal to negative one. Finally, dividing both sides by two, we have 𝑥 is equal to negative one-half.
We can give the answer as a solution set either fractionally or as decimals. Using the decimal form, we can say that the solution to two 𝑥 minus one to the power of six equals 64 is the set containing 1.5 and negative 0.5.
