Question

How do you simplify `(3sqrt7 - 6sqrt5)(4sqrt7 + 8sqrt5)`?

Answer 1

`(3sqrt7-6sqrt5)(4sqrt7+8sqrt5)=-156`

Explanation:

Simplifying would mean getting rid of as many square roots as possible. We should, therefore, multiply it out.

First we note that 3 is common in the first parenthesis, and 4 is common in the second. Therefore:

`(3sqrt7-6sqrt5)(4sqrt7+8sqrt5)`
`=3(sqrt7-2sqrt5)*4(sqrt7+2sqrt5)`

You should know that `(a-b)(a+b)=a^2-b^2`,
which means that `(sqrt7-2sqrt5)(sqrt7+2sqrt5)=7-2^2*5=-13`

Therefore `=3(sqrt7-2sqrt5)*4(sqrt7+2sqrt5)=-3*4*13=-156`

Ergo `(3sqrt7-6sqrt5)(4sqrt7+8sqrt5)=-156`