Dripread - Book

Whenever we wish to solve an expression, we should first rearrange it to make it as simple as possible, then manipulate it. If we try to manipulate expressions in a complex form, we increase the possibility of making mistakes or becoming confused. Once the expression has been simplified, we can either solve it or state it in its most simplified form.

Sequence of arithmetical calculations

Brackets are used to avoid any ambiguity in a calculation. Any calculations inside a bracket are usually carried out first.

For example, 2 × (7 + 4) = 2 × 11 = 22 or 2 × 7 + 2 = 4 = 22

Without brackets, 2 × 7 + 4 could mean either (2 + 7) + 4 or 2 × (7 + 4) but see below for conventions on which operations are done first.

Similarly, 2 − (8 − 9) is not the same as 2 − 8 − 9

2 − (8 − 9) = 2 − (−1) = 2 + 1 = 3 or 2 − 8 + 9 = 3

whereas 2 − 8 − 9 = 2 − 17 = −15

Algebra has rules and conventions in the way that it operates. In other words, there are things that you have to do (rules) and things that mean that mathematics is presented consistently (conventions). We saw examples of rules in Section 3 on indices. Conventionally, when there are no brackets, multiplication and division are done before addition and subtraction; most calculators will do this automatically.

For example:

Other useful conventions are:

writing 4n instead of 4 × n; 4mn is 4 × m × n
writing 4n rather than n4
writing the product of multiplying x by itself as x² instead of xx
writing x rather than 1x.

We will now look at some examples of simplifying expressions.

Example 2

Simplify:

Discussion

There are three kinds of terms in this expression. Simple numbers like 4, terms like 12t and terms like 13t².

Two useful rules follow:

when adding like terms, add the coefficients; for example, 2x + 5x = 7x
avoid mixing powers of the same variable ? you can't add x + 2x² to give a single term, as they represent different numbers; for example, when x = 2, x² = 4.

By convention, the terms with the highest power go at the start of the expression.

So, start by collecting all the terms that relate to t and t² together, and begin the statement with the terms that relate to t².

Now, we will add these terms together, and simplify as far as possible.

Activity 5

Simplify the following

Answer

Discussion

(but do try these before you look at the answers).

View larger image

We can now go on to discuss further the use of brackets in mathematics. As you have seen, brackets are used to indicate the order in which a numerical calculation should be carried out. This is also true when you are using symbols.

So, for example, 2(2n + 5) = 4n + 10

As a convention, 2(2n + 5) is preferable to (2n + 5)2, although, since it doesn't matter what order you multiply two numbers in, they are equivalent.

To multiply out brackets, the critical factor is that all of the expression inside the brackets is multiplied by the figure outside them.

So, for example, x − 2(x − 3) = x − 2 × x − 2 × −3 = x − 2x + 6 = −x + 6

If we are multiplying two expressions within brackets together, such as (x + 2)(x−1), we can approach this in several ways. The most common is to split the first bracket out, so