Question

How do you simplify `(12x^6)/(6x^ -2)`?

Answer 1

`(12x^6)/(2x^(-2)) = 2x^8`

Explanation given in detail. The method should help you deal with any question of this type.

Explanation:

Splitting this into 2 parts; numbers and variables

`color(blue)("Part 1: The numbers") color(white)(".....") 12/6`

Divide top and bottom by 6 as `12 -: 6` gives a whole number answer.

`(12 -: 6)/(6 -:6) = 2/1 = 2`
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`color(blue)("Part 2 : The variables") color(white)(".....") (x^6)/(x^-2)`

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`color(brown)("To explain what is happening: ")`

Suppose we had a different variable, say `z`.

If this is written as `z^(-2)` then it is another way of writing `1/(z^2)`
If this is written as `1/(z^(-2))` then it is another way of writing `z^2`

So to summarise: If written this way in the numerator it is actually the denominator. On the other hand, if it is written this way in the denominator it is actually the numerator.

`color(green)("You 'move it' to the other side of the dividing line.")`
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Write `x^6/(x^(-2)) " as " x^6 xx x^2`

But `x^6 xx x^2` is the same value as `x^((6+2)) = x^8`

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`color(blue)("Putting it all together")`

`(12x^6)/(2x^(-2)) = 2x^8`

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Another way of dealing with `1/x^(-2)`

Multiply by 1 in the form of`color(white)(..)x^2/x^2` giving:

`1/x^(-2) xx x^2/x^2`

`=(1xx x^2)/(x^(-2)xxx^(2))`

`= x^2/(x^((-2+2)))`

`x^2/x^0`

But `x^0=1` giving

`x^2/1 = x^2`