Video: Finding the 𝑛th Root of a Number

Complete the following: ∜16 = √_.

02:04

Video Transcript

Complete the following: The fourth root of 16 equals the square root of blank.

In this question, on the right-hand side, we have a value which is taken to the square root, and we know that it’s equal to the fourth root of 16. Let’s therefore begin by working out the fourth root of 16. But first, a note, this is not the same as four multiplied by the square root of 16. We can tell the difference by looking at the size of the digits. For example, when we’re working out four multiplied by the square root of 16, the four and the 16 should be the same size. However, when we’re taking an 𝑛th root, for example, the fourth root here, the four will be smaller and usually set within the root sign.

So in order to actually calculate the fourth root of 16, it’s equivalent to thinking that there’s a value 𝑥 written to the fourth power, which would give us an answer of 16. 𝑥 to the fourth power is 𝑥 written four times and multiplied together. If we tried the value of 𝑥 equals two, we’d need to work out two times two times two times two, and two times two is four. Four multiplied by the third two would give us eight. And eight multiplied by the final two would give us 16. Therefore, two to the fourth power is 16. And importantly for us, that means that the fourth root of 16 is two.

However, when we look back at the original question, we can’t just fill in the value two onto the blank because it’s this whole expression on the right-hand side that needs to be equal to two. Therefore, we need to think about what value would give us an answer of two when we take the square root of it. Well, two squared would give us four. And therefore, the square root of four would give us two. We can therefore give our answer that the missing value is four since the fourth root of 16 is equal to the square root of four.

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