Rekenen met variabelen *: Variabelen *
Order of operations
When calculating with numbers, keep in mind the order of operations. These determine the order in which the operations are to be carried out.
Order of operations
When interpreting an expression consisting of additions, subtractions, multiplications, divisions and exponentiation, we have to take into account the following rules:
- additions and subtractions are performed according to the order the operations are found in the expression at hand, from left to right
- multiplication and divisions are performed according to the order the operations are found in the expression at hand, from left to right
- multiplication and division have precedence over additions and subtractions
- exponentiation has precedence over multiplication and division
- exponentiation is performed according to the opposite order the operations are found in, from right to left
You can change the order using brackets. The operation inside the brackets always has precedence.
A mnemonic to remember the order of operation is
"Please Excuse My Dear Aunt Sally"
The bold letters give the interpretation: first remove the parenthesis (brackets), then exponentiation, then multiplication and division, followed by addition and subtraction.
Indeed,\[ \left(\frac{12}{2}\right)^2 = 6^2=36\tiny.\]
Algebraic expressions
When you replace numbers by variables in a mathematical expression, you create an algebraic expression. The same order of operation as above is used when calculating with variables.
When you replace variables by numbers, the result of this substitution can be interpreted in the same way as when you calculate with numbers. Thus, you can calculate with variables as with numbers; for example \(a+b\) can be called the sum of \(a\) and \(b\).
The number in front of a product of variables, is called a coefficient. For instance, #3# is the coefficient of #a^2b# in #a^2+3a^2b#.
The following notation rules are used when calculating with variables.
- When multiplying, the times sign \(\times\) is often substituted by a dot \(a\cdot b\); sometimes it is completely omitted: \(a\,b\). When you fill in your answer in the appropriate answer field, it is necessary to separate #a# and #b# with #\cdot# in order to avoid confusion with a variable consisting of two letters (with the name #ab#).
- The order of the terms in a multiplication is of no influence to the result; for instance, #a\cdot b^3=b\cdot a\cdot b\cdot b#. In the mixed forms of numbers and variables, for example \(2ab^3\), it is common to put the coefficient first: so \(2ab^3\) rather than \(a2b^3\). Here, #2# is the coefficient of #ab^3#.
- If the notation is open for interpretation or is simply hard to read, one uses brackets to indicate precedence and a dot to indicate multiplication: according to the rules \(2ab^3\) is equal to \(2\cdot a\cdot\ b^3\) and not equal to \(2(ab)^3\) or \((2ab)^3\).
A space in the answer field is often not recognized by the computer. So, if you mean to write "#a# times #b#", you should enter #a\cdot b#. The expression #a\, b# will be interpreted as the variable #ab# rather than the product #a\cdot b#.
Substituting #a=3# in the expression \[2\cdot a^3+2\cdot a^2\] gives \[2\cdot 3^3+2\cdot 3^2\tiny.\] Further calculations lead to the result #72#.
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