Brackets

Preknowledge

List here

Brackets in mathematics are used to show the order in which you must do things. This is important as you can get different answers depending on the order in which you do things. For example

[math](5 \times 5) + 20 = 45[/math]

whereas

[math]5 \times (5 + 20) = 125[/math]

If there are no brackets, you should always do multiplications and divisions first and then additions and subtractions3. You can always put your own brackets into equations using this rule to make things easier for yourself, for example:

[math]a \times b + c \div d = (a \times b) + (c \div d)[/math]

[math]5 \times 5 + 20 \div 4 = (5 \times 5) + (20 \div 4)[/math]

If you see a multiplication outside a bracket like this

[math]a(b + c)[/math]

[math]3(4 - 3)[/math]

then it means you have to multiply each part inside the bracket by the number outside

[math]a(b + c) = ab + ac[/math]

[math]3(4 - 3) = 3 \times 4 - 3 \times 3 - 12 - 9 = 3[/math]

unless you can simplify everything inside the bracket into a single term. In fact, in the above example, it would have been smarter to have done this
[math]3(4 - 3) = 3 \times (1) = 3[/math]

It can happen with letters too [math]3(4a - 3a) = 3 \times (a) = 3a[/math]

Review Quiz

1. Which is the correct way to solve this problem? x(y + z) (a) xy + xz → Correct! (b) (x + y) + (x + z) → Incorrect. (c) xy + z → Incorrect. (d) x + (yz) → Incorrect. Your score is 0 / 0

Extension exercise

Distributivity The fact that [math]a(b + c) = ab + ac[/math] is known as the distributive property. If there are two brackets multiplied by each other, then you can do it one step at a time: [math](a + b)(c + d)[/math] [math]= a(c + d) + b(c + d)[/math] [math]= ac + ad + bc + bd[/math] [math](a + 3)(4 + d)[/math] [math]= a(4 + d) + 3(4 + d)[/math] [math]= 4a + ad + 12 + 3d[/math]