## 4. Expressions

Expressions are ways of writing the interaction between different variables, or a variable and one or more parameters. In the relationship E = m.C^{2}, 'm.C^{2}' is an expression that combines a variable (mass) and a parameter (the velocity of light).

Expressions should be written in a way that there is only one way to evaluate them. Where expressions are complex, brackets are used to indicate how the expression should be evaluated. The acronym BODMAS is used as a guide to evaluating expressions:

BRACKETS: | Evaluate the contents of the brackets first, working from the innermost (lowest in the hierarchy) to the outermost (highest in the hierarchy) |

'ORDER': | Evaluate powers |

DIVISION AND MULTIPLICATION | Take priority over … |

ADDITION AND SUBTRACTION | … which are evaluated last |

Conventionally, when the above priorities are followed, the expression is then evaluated from left to right. This is what computers and calculators do automatically, although you might find yourself selecting any part of a complex expression in order to start to simplify it. Very complex expressions may need to be broken up into separate sub-expressions that are evaluated separately.

The following table shows expressions including the variable 'a' and a parameter represented by the symbol 'b'. For clarity, multiplication is shown using a full-stop (period) as a separator, but this may not always be shown in expressions, especially if the multiplier is a number rather than a symbol. The multiplication sign '×' is not normally used in expressions, and lower-case x should never be used to represent this symbol. Division is written using '/' (backslash or solidus), and the division sign '÷' is not used.

a | A variable |

3 . a | The variable a multiplied by 3 (also written simply as '3a') |

b . a | The variable a multiplied by b (also called the 'product' of a and b) |

a / b | The variable a divided by b |

b . a^{2} |
The variable a-squared, then multiplied by b |

(b . a)^{2} |
The variable a multiplied by b, with the result squared (note the effect that the brackets have) |

(b . a^{2})^{3} |
The variable a-squared multiplied by b, with the result raised to the power of three |

b . a + a^{2} |
The product of a and b, added to a-squared |

b . (a + a^{2}) |
The sum of a and a-squared, with the result multipled by b (again, note the effects of the brackets) |

a + a^{2} / b |
The variable a-squared divided by a, then added to a |

((a + a^{2}) / b)^{0.5} |
The square-root of the previous expression (inner brackets are evaluated first) |

Operations on simple expressions are demonstrated in the 'Equations and how to solve them' resource on the NuMBerS site.