### Video Transcript

If point π΄ on the number line represents the cube root of negative 27 and point π΅ represents the square root of nine, which of the following line segments has greater length? Option (A) the line segment between π΄ and π΅. Option (B) the line segment between πΆ and π΅. Or is it option (C) the line segment between π΄ and πΆ?

In this question, we are given a point πΆ on a number line at zero. And we are told that points π΄ and π΅ should be the cube root of negative 27 and the square root of nine on the number line, respectively. We need to use this to determine which of three line segments is the longest.

To answer this question, we can start by finding where points π΄ and π΅ should be on the number line. We can start by recalling that the square root of nine is the nonnegative number whose square is nine. We know that three squared is nine. So, the square root of nine is three. This allows us to mark point π΅ on the number line at three. In the same way, we can recall that the cube root of negative 27 will be the number whose cube is negative 27.

We can note that negative three cubed is equal to negative 27. So, the cube root of negative 27 is negative three. This then allows us to mark point π΄ on the number line at negative three. This then allows us to find the lengths of the three line segments using the number line. We have that π΄π΅ is equal to six, πΆπ΅ is equal to three, and π΄πΆ is equal to three.

It is worth noting that we can answer this question without using a number line. If points π₯ and π¦ represent values π₯ and π¦ on a number line, then the length of line segment π₯π¦ is equal to the absolute value of π₯ minus π¦. This then allows us to calculate the lengths algebraically. For instance, π΄π΅ is equal to the absolute value of negative three minus three, which we can calculate is equal to six. In either case, we were able to show, of the three line segments, the line segment π΄π΅ has the greatest length, which is option (A).