Narrator: The next rearrangement looks more complicated. Although you won't meet this equation in S104, it's been included to show how you can use the same principles to rearrange any equation, however horrible it might look.

Shelagh: Right, our next equation is describing the speed v s of a wave going through a rock of rigidity modulus mu and density rho.

Alison: [Laughter]

Shelagh: [Laughter] And I would like you to find, er, an equation for rho.

Alison: Right.

Shelagh: Um, the thing to note is that I've drawn this square root sign so that it goes around the whole of the fraction.

Alison: Yeah.

Shelagh: You could put brackets around the fraction but, because it's drawn that way, it's not necessary.

Alison: Right.

Shelagh: Where do you think you might start with this one?

Alison: Well, um, the rho is on the bottom on the right-hand side and I need it on the top on the left-hand side but there's a great big square root symbol in the way and I don't know what to do with it.

Shelagh: Well, what's the opposite operation to taking a square root?

Alison: Squaring.

Shelagh: Yeah. So ...

Alison: So if I square both sides of the equation. I'll get rid of the square root.

Shelagh: You will.

Alison: So, um, if I do v s squared, er, is equal to mu over rho because I've cancelled that square root out by squaring.

Shelagh: Yes.

Alison: Yes.

Shelagh: But now you've still got ...

Alison: ... rho on the bottom

Shelagh: ... on the bottom on the right.

Alison: Okay, lovely, so, um ...

Shelagh: So how can you put rho on the top on the left?

Alison: I need to multiply both sides by rho.

Shelagh: You do.

Alison: So I'll just do that and that and then the rhos on the right-hand side cancel out and I'm left with v s squared rho is equal to mu.

Shelagh: Okay. So where were we ...

Alison: I've forgotten what I was supposed to do. [Laughter].

Shelagh: [Laughter]. Where were we heading? We were trying to find rho.

Alison: Okay. Right.

Shelagh: And now we've got rho multiplied by vs squared. So you know what to do with that?

Alison: Yes, divide both sides by v s squared, divided by v s squared so the v s squareds cancel on the left-hand side and I'm left with rho is equal to mu over v s squared.

Shelagh: You got there, well done. [Laughter].

Narrator: As the equations become more complicated, it becomes more important to tackle them in several small steps, as Alison did in this example. Also, we have now extended the general rule, that to undo any operation, you should do the opposite. In this case Alison needed to get rid of a square root, so she remembered that squaring something and finding a square root are opposite operations. So to get rid of the square root, she squared both sides of the equation. Before attempting the final problem you may like to take a break and try parts (a) and (b) of Question 3.3, immediately after this activity in the course book.