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Where am I going wrong
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Hi u/scillywoba,

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When you multiply something into a sum, you need to multiply with every term. Like this:   
   
(a + b)x = ax + bx   
   
Similarly for when you divide. Thus in the division on the top row can't cross out the x's like that, because then you skip the division of the second term.
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You can't cancel terms that are inside a sum.

You can only cancel terms that are factors (part of a product).

(x^3 + ux)/x^3 does not have a product in numerator.

But you can factorize/factorise the numerator to make it a product of x and (x^2 + u).

= (x * (x^2 + u)) / (x * x^2 )

= (~~x~~ * (x^2 + u)) / (~~x~~ * x^2 )

= (x^2 + u) / x^2

The common factor x in each of the terms in the numerator can be pulled out as a factor using distributive property. Then you do see factor x in top and bottom that cancels, because x/x = 1.

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A simpler example:

(5x + 3)/x is equal to 5x/x + 3/x = 5/1 + 3/x = 5 + 3/x.

You can't say (5x + 3)/x = (5~~x~~ + 3)/~~x~~ = (5(1) + 3)/1 = 8
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As someone else has said, you can’t cancel part of addition or subtraction. If you have (2+7)/2 can you cancel the 2’s leaving the answer being 7? No.

So on your first line you can’t cancel the x^3 .

Not sure why you are multiplying the top and bottom by x^3 on the second line? Where does the x^3 on the bottom go?
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Top equation should be:

(x^3 + ax)/x^3 = 1 + (a/x^2 )

Basically, as others said, you can only cancel factors, not addition.
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This is hideous to look at
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(x^3 + ux)/x^3 =/= ux

Because if you factor the numerator, you get:

x(x^2 + u)

As there is only one x in both terms in the numerator.

So you’d really get (x^2 + u)/x^2, which can be rewritten as 1 + u / x^2

Realistically this is about as simplified as you can get without any other information

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