(x^(2)-y^(2))/(x-y)

x^(2) \- y^(2) is the 'difference of two squares'. If you can spot that, we know that means it factorises as (x + y)(x - y)... cross multiply out the brackets if you need to convince yourself that works.

Next, we can divide both top and bottom of the fraction to simplify, so divide both by the common factor of (x - y) and we're done!
x = 6

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y = 4

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x + y = 6 + 4 = 10

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x - y = 6 - 4 = 2

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x^(2) = 36

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y^(2) = 16

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x^(2) - y^(2)

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(x - y) + (x - y) + (x - y) + (x - y)

\+ (x - y) + (x - y) + (x - y) + (x - y)

\+ (x - y)^(2)

= y(x-y) + y(x-y) + (x-y)^(2)

= 2y(x-y) + (x-y)^(2)

= (y + y + (x-y))(x-y)

= (x + y - y + y)(x-y)

= (x + y)(x - y)

This proves that

x^(2) - y^(2) = (x + y)(x - y).

Now the fraction

(x^(2) - y^(2))/(x - y)

can be simplified by cancelling the (x - y) on numerator and denominator

(x + y)(x - y)/(x - y)

= (x + y)