Question Video: Simplifying and Solving Equations Involving 𝑛th Roots Mathematics

Video Transcript

Find the value, or values, of 𝑥 if 12𝑥 to the fifth power equals 384.

We’re going to solve this equation for 𝑥 by performing a series of inverse operations. We notice that 𝑥 to the fifth power is being multiplied by 12. So we’re going to begin by dividing both sides of this equation by 12. 384 divided by 12 is 32. So we get 𝑥 to the fifth power equals 32. Then, to solve this equation, we recall what we know about even and odd roots. Specifically, given the equation 𝑦 to the 𝑛th power equals 𝑥, if 𝑛 is odd, then there is exactly one solution to this equation, 𝑦 equals the 𝑛th root of 𝑥.

Now, since our equation is in terms of 𝑥, we’re going to rewrite this slightly. So if 𝑥 to the 𝑛th power equals 𝑦, then 𝑥 is equal to the 𝑛th root of 𝑦. Now, in this case, 𝑛 is odd; it’s equal to five. And so there is just one solution to this equation. It’s 𝑥 equals the fifth root of 32. But of course, the fifth root of 32 is simply equal to two. And so, the solution is 𝑥 equals two.