Question

How do you multiply `(sqrt5-sqrt6)(sqrt5+sqrt2)`?

Answer 1

See a solution process below:

Explanation:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

`(color(red)(sqrt(5)) - color(red)(sqrt(6)))(color(blue)(sqrt(5)) + color(blue)(sqrt(2)))` becomes:

`(color(red)(sqrt(5)) xx color(blue)(sqrt(5))) + (color(red)(sqrt(5)) xx color(blue)(sqrt(2))) - (color(red)(sqrt(6)) xx color(blue)(sqrt(5))) - (color(red)(sqrt(6)) xx color(blue)(sqrt(2)))`

We can now use this rule for radicals to multiply the radicals:

`sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))`

`sqrt(color(red)(5) xx color(blue)(5)) + sqrt(color(red)(5) xx color(blue)(2)) - sqrt(color(red)(6) xx color(blue)(5)) - sqrt(color(red)(6) xx color(blue)(2))`

`sqrt(25) + sqrt(10) - sqrt(30) - sqrt(12)`

`5 + sqrt(10) - sqrt(30) - sqrt(12)`

We can then use this rule for radicals to simplify the term on the right:

`sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))`

`5 + sqrt(10) - sqrt(30) - sqrt(12)`

`5 + sqrt(10) - sqrt(30) - sqrt(color(red)(4) * color(blue)(3))`

`5 + sqrt(10) - sqrt(30) - (sqrt(color(red)(4)) * sqrt(color(blue)(3)))`

`5 + sqrt(10) - sqrt(30) - 2sqrt(3)`