multiply each # by its conjugate x+√y 2-√3  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that changing the sign between the two terms yields the conjugate of the number. Hence, supposing that the number is of form `a + sqrt b` , thus the conjugate is `a - sqrt b` .

Hence, reasoning by analogy, the conjugate of `x+sqrt y`  is `x - sqrt y` .

Performing the multiplication between the number and its conjugate yields a diference of two squares, such that:

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

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

Considering the number `2 - sqrt 3` , hence its conjugate is `2 + sqrt 3`  and performing the multiplication yields:

`(2 - sqrt 3)(2+ sqrt 3) = 2^2 - (sqrt3)^2 = 4 - 3 =1`

Hence, performing the multiplications of the given numbers by its conjugates yields `(x+sqrt y)(x-sqrt y) = x^2 - y`  and `(2 - sqrt 3)(2 + sqrt 3) = 1` .

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