Evaluate the fourth root of 81.
Before we begin answering this question, it’s important to notice the distinction between this root sign with a small four or a large four. What we have here in the question is the fourth root of 81. This second value, however, would indicate four multiplied by the square root of 81. We must remember that if we’re handwriting these to make sure that when we’re writing the fourth root or something like it to make the value in the root smaller. Four multiplied by the square root of 81 is not what we want to calculate here.
In order to calculate the fourth root of 81, we’re really asking, what value of 𝑦, for example, to the fourth power would give us a value of 81? When we’re finding a number to the fourth power, that means we have four of that number multiplied together. We can often solve these through a series of trial and improvement. If we took the value of 𝑦 equals one, we know that one times one times one times one would only give us one and not 81. So, let’s try a value of two. Two times two would give us four. Four times two would give us eight. And eight times two would give us 16. But 16 isn’t the value that we’re looking for.
Next, we can try the value of three. To work out three times three times three times three, we can do it in several different methods. We could work out the first three times three, which is nine, and then we could multiply it by three and then multiply it by another three. Or we could notice that there’s another three times three in there, which gives us nine. And nine times nine gives us a value of 81. This means that we’ve found the value that we were looking for. Since we now know that three to the fourth power is 81, then that means that the fourth root of 81 is three. And so, three is our answer.