Question

How do you evaluate 4√112 +5√56 - 9√126?

Answer 1

`16sqrt(7)-17sqrt(14)`

Explanation:

Note that

`112=7*16`

`56=4*14`

`126=9*14`

and

`sqrt(ab)=sqrt(a)*sqrt(b)` if `a,b>=0`
so we get

`16sqrt(17)+10sqrt(14)-27sqrt(14)`

combining like Terms

`16sqrt(7)-17sqrt(14)`

Answer 2

`16sqrt7-17sqrt14`

Explanation:

Write each radicand (the number under the root) as a product of its factors. Try to use perfect squares wherever possible.

`4sqrt112+5sqrt56-9sqrt126`

`= 4sqrt(16xx7) +5sqrt(4xx2xx7) -9sqrt(9xx14)`

Find any roots possible:

`=4xx4xxsqrt7 +5 xx2 sqrt(2xx7)-9xx3sqrt(2xx7)`

`=16sqrt7 +10sqrt14-27sqrt14`

`=16sqrt7-17sqrt14" "larr` this can be the answer

Or, we can try to improve it a bit

`=16sqrt7-17sqrt7sqrt2`

This can be factored to give:

`sqrt7(16-17sqrt2)`

However, this answer is no better than the first.

Answer 3

`16sqrt7-17sqrt14`

Explanation:

We have the following:

`4sqrt112+5sqrt56-9sqrt126`

In each of our terms, we can rewrite the radicals in such a way where we can factor out a perfect square:

`color(blue)(4sqrt(16*7))+color(purple)(5sqrt(4*14))-color(steelblue)(9sqrt(9*14))`

This business simplifies to

`color(blue)(16sqrt7)+color(purple)(10sqrt14)-color(steelblue)(27sqrt14)`

Since the latter two terms have a `sqrt14` in common, we can simplify those to get

`color(blue)(16sqrt7)-color(red)(17sqrt14)`

Since the radicals have no perfect square factors, we are done!

Hope this helps!

Answer 4

`16sqrt7-17sqrt14`

Explanation:

`4sqrt112+5 sqrt 56-9 sqrt 126`

`:.=4 sqrt 7*sqrt 16+5 sqrt 7* sqrt 8-9 sqrt(2*3*3*7)`

`:.=4*4 sqrt 7+5 *sqrt 7*sqrt 8-27*sqrt 2*sqrt 7`

`:.=sqrt 7(16+5 sqrt(2*2*2)-27 sqrt 2)`

`:.=sqrt 7(16+10 sqrt 2-27 sqrt 2)`

`:.=sqrt 7(16+sqrt 2(10-27))`

`:.=sqrt 7(16+sqrt 2(-17))`

`:.=sqrt 7(16-17 sqrt 2)`

`:.=16 *sqrt 7-17 sqrt 2*sqrt 7`

`:.=16sqrt 7-17 sqrt 14`