Question Video: Arranging Numbers in Descending Order Involving Square and Cube Roots Mathematics • Second Year of Preparatory School

Video Transcript

Arrange the following numbers in descending order: the square root of 33, the negative square root of 22, the square root of 41, three, zero, and the cube root of negative 125.

In order to be able to arrange these numbers, we have to make use of some of the properties of square roots.

For example, if the square of a number 𝑎 is less than another number 𝑏 and both numbers are positive, then 𝑎 is less than the square root of 𝑏. So for instance, let us consider the numbers root 33 and three. Then we can observe that since three squared, which is nine, is less than 33, then we can conclude that three is less than root 33.

Another useful rule that we can use is that if 𝑎 is less than 𝑏, then root 𝑎 is also less than root 𝑏. We can use this property on root 33 and root 41. That is, since 33 is less than 41, then root 33 is less than root 41. If we were to put these three numbers on a number line, they would look roughly like this. We also know that zero is less than three. If we consider the remaining two numbers, we can see that they will both be negative. Therefore, they will be somewhere on this line to the left of zero.

To find the exact ordering of these two numbers, we should note that the five cubed is five times five times five, which is 125. Therefore, negative five cubed is negative 125. Then, if we cube root both sides, we have that negative five is equal to the cube root of negative 125.

We also know that root 22 is less than root 25. Therefore, by flipping the inequality, negative root 22 will be greater than negative root 25. But since root 25 is equal to five, we have that negative root 22 is greater than negative five.

Let us also not forget that negative five is equal to the negative cube root of 125. So the negative root of 22 is greater than the cube root of negative 125. We can thus append these numbers to our number line.

Therefore, by arranging the numbers in descending order, we have the square root of 41, the square root of 33, three, zero, the negative square root of 22, and the cube root of negative 125.