Rearranging Equations – Expanding Brackets

Brackets in equations show that everything in the bracket is subject to the action on the outside of the bracket. This means we need to perform an operation to get the bracket out of the way before we isolate the terms within, using inverse operations.

Let’s work through an example.

In the equation below, we need to isolate h:

rho space equals space 2 space left parenthesis w italic space italic plus italic space h right parenthesis

Before we can isolate h we need to get rid of the brackets. To do this, we need to multiply both w and h  by 2 to get:

rho space equals space 2 w space italic plus italic space italic 2 h

Now, as 2w + 2h is the same as 2h + 2w we can shift them around to make:

rho space equals space 2 h space italic plus italic space italic 2 w

We can then subtract 2w from both sides giving us:

rho space minus space 2 w space italic equals italic space italic 2 h

And then divide both sides by 2 to get h on its own:

h space equals space fraction numerator rho space minus space 2 w over denominator 2 end fraction

It is important to note that you can’t do this if you are squaring (or exponentiating) the terms in the brackets because:

left parenthesis 2 space plus space 4 space plus space 6 right parenthesis squared

Does not equal:

2 squared space plus space 4 squared space plus space 6 squared

In this situation, we would just take a square root of both sides of the equation to cancel the squaring (exponentiation) and dispense with the brackets.

To help reinforce what we have just learned, on the next page there are some practice questions. Answers and explanations can be revealed by clicking on show answer.

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