Question

What is the following product? (StartRoot 12 EndRoot + StartRoot 6 EndRoot) (StartRoot 6 EndRoot minus StartRoot 10 EndRoot)

Options: A. 6 StartRoot 2 EndRoot minus 2 StartRoot 30 EndRoot + 6 minus 2 StartRoot 15
B. EndRoot 6 StartRoot 3 EndRoot minus 6 StartRoot 5
C. EndRoot 3 StartRoot 2 EndRoot minus StartRoot 22 EndRoot + 2 StartRoot 3 EndRoot minus 4
D. 2 StartRoot 3 EndRoot + 6 minus 2 StartRoot 15 EndRoot

Answer 1

The product results in:
`6\,√(2) -2\,√(30) +6-2\,√(15)`, which agrees with answer A of the given choices.

Explanation:

We need to apply distributive property for the product of two expressions each consisting of two terms, and also use the properties of products of radicals of the same root:

`(√(12) +√(6) )(√(6) -√(10) )=\\=√(12*6) -√(12*10) +√(6*6) -√(6*10) =\\=√(72) -√(120) +√(36) -√(60)`

and now, we extract as many factors we can from the roots to reduce them:

`√(72) -√(120) +√(36) -√(60) =\\=6\,√(2) -2\,√(30) +6-2\,√(15)`