Express the 11th root of 𝑥 to the seventh in the form 𝑥 to the 𝑛.
One of the tools that we’ll need to solve this problem is the rational exponent rule which tells us that 𝑥 to the 𝑚 over 𝑛 equals the 𝑛 root of 𝑥 to the 𝑚. That sounds really wordy, but we’ll break it down.
In our problem, we’re given something that looks like this and asked to convert it to something that looks like this. Let’s identify what would be the 𝑛 and 𝑚 in our case. The root that we’re dealing with, the 𝑛 in our problem is 11. The exponent, what the 𝑥 is being taken to the power of, in our case is seven. Our 𝑚 would be seven. And both the rule and our problem is working with 𝑥, so the 𝑥 remains the same.
Now we need to plug in what we know. Our problem should be 𝑥 to the 𝑚 over 𝑛. Our 𝑚 equals seven, and our 𝑛 equals 11.
The rational form of the expression we were given is 𝑥 to the seven over 11th power.