Firstly you have to know what (a+b+c)(d+e+f) is. You can expand this expression using a rectangle:

So you know that:

Next you’d have to multiply (a+b+c)(d+e+f) by (g+h+i) using another rectangle:

And from here you’d figure out that:

An **even function **exists when for all values of x.

The graph of an **even function **must be symmetrical about the y-axis.

Examples of **even functions **have been listed below:

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An **odd function** exists when for all values of x.

Graphs of **odd functions **should have 180 degrees rotational symmetry about the origin (0,0).

Examples of **odd functions **are listed below:

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Most functions are neither even or odd.

An example of a function that is neither even or odd would be:

**Commutative** means that the order of elements contained within an expression do not alter a certain mathematical result.

For example:

Knowing this, you can say that:

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**Associative **means that you can alter the grouping of certain elements in an expression without changing its result.

Knowing this you could say that:

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Warning:

**Commutative** and **Associative** manipulation should not be used when subtracting and dividing.

First Scenario:

Second Scenario:

Notice that what we’re ultimately doing in both cases is multiplying the surd within a fraction by 1. When the value of the numerator is exactly the same as the value of the denominator in a fraction, what you have is 1.

You should know that 1/1, 2/2, 3/3, (a+b)/(a+b) are all equal to 1.

In order to multiply surds, you should first know these rules:

You should also know that:

So, knowing these rules, what would you get if you multiplied: ?

Well, .

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Now how about ?

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What about ?

Prove that:

Firstly:

Also, remember that:

So:

— USEFUL FORMULAS:

Firstly, you need to know what the product rule is:

Then…

Now integrate each term with respect to x:

Leaving:

Prove that: