Where am I going wrong

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.
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
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?
Top equation should be:

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

Basically, as others said, you can only cancel factors, not addition.
This is hideous to look at
(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