Question

How do you simplify `(5sqrt2+3sqrt5)(2sqrt10-5)`?

Answer 1

`=5(sqrt5 +sqrt2)`

Explanation:

You can multiply the brackets using the distributive law, in exactly the same way as in cases such as:

`(x+3)(x-4) = x^2 -4x +3x-12`

Each term in the first bracket must be multiplied by each term is the second bracket.

`(color(red)(5sqrt2) +color(blue)(3sqrt5))(2sqrt10-5)`

`=color(red)(5sqrt2)(2sqrt10-5) +color(blue)(3sqrt5)(2sqrt10-5)`

`=10sqrt20-25sqrt2+6sqrt50-15sqrt5`

Now find factors for the roots, using squares where possible:

`=10sqrt(color(magenta)(4)xx5)-25sqrt2+6sqrt((color(lime)25xx2)-15sqrt5`

`=10xxcolor(magenta)(2)sqrt5-25sqrt2+6 xxcolor(lime)5sqrt2 -15sqrt5`

`=20sqrt5 -15sqrt5+30sqrt2-25sqrt2`

`=5sqrt5+5sqrt2`

`=5(sqrt5 +sqrt2)`

Answer 2

`5(sqrt2+sqrt5)`

Explanation:

`(5sqrt2+3sqrt5)(2sqrt10-5)`

`color(white)(aaaaaaaaaaaaa)` `5sqrt2+3sqrt5`
`color(white)(aaaaaaaaaaaaa)` `2sqrt10-5`
`color(white)(aaaaaaaaaaaaa)` `-----`
`color(white)(aaaaaaaaaaaaa)` `10sqrt2sqrt10+6sqrt5sqrt10`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa)` `-25sqrt2-15sqrt5`
`color(white)(aaaaaaaaaaaaa)` `-------------`
`color(white)(aaaaaaaaaaaa)` `color(blue)(10sqrt2sqrt10+6sqrt5sqrt10-25sqrt2-15sqrt5`

`:.=10sqrt2 xx sqrt2 xx sqrt5+6sqrt5 xx sqrt2 xx sqrt5-25sqrt2-15sqrt5`

`:.=10 xx 2sqrt5+6 xx 5 xx sqrt2-25sqrt2-15sqrt5`

`:.=20sqrt5+30sqrt2-25sqrt2-15sqrt5`

`:.=20sqrt5-15sqrt5+30sqrt2-25sqrt2`

`:.=5sqrt5+5sqrt2`

`:.color(blue)(=5(sqrt2+sqrt5)`