Question

How do you show that if you multiply `asqrtb+csqrtf` and `asqrtb-csqrtf`, the product has no radicals?

Answer 1

`(asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f)) = a^2b-c^2f`

Explanation:

Note that:

`(A+B)(A-B) = A^2-B^2`

So we find:

`(asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f)) = (asqrt(b))^2-(csqrt(f))^2`

`color(white)((asqrt(b)+csqrt(f))(asqrt(b)-csqrt(f))) = a^2b-c^2f`