Question

How do you evaluate `-\sqrt { 18} - 3\sqrt { 6} + 3\sqrt { 24}`?

Answer 1

See the solution process below:

Explanation:

We can use this rule for radicals to rewrite the term on the left of the expression:

`sqrt(a * b) = sqrt(a) * sqrt(b)`

`-sqrt(18) - 3sqrt(6) + 3sqrt(24) => -sqrt(9 * 2) - 3sqrt(6) + 3sqrt(24) =>`

`-(sqrt(9) * sqrt(2)) - 3sqrt(6) + 3sqrt(24) =>`

`-3sqrt(2) - 3sqrt(6) + 3sqrt(24)`

We can next factor out a `3` from each term in the expression:

`(3 * -sqrt(2)) + (3 * -sqrt(6)) + (3 * sqrt(24)) =>`

`3(-sqrt(2) - sqrt(6) + sqrt(24))`

Now, rewrite `sqrt(24)` to give:

`3(-sqrt(2) - sqrt(6) + sqrt(24)) => 3(-sqrt(2) - sqrt(6) + sqrt(4 * 6)) =>`

`3(-sqrt(2) - sqrt(6) + (sqrt(4) * sqrt(6))) => 3(-sqrt(2) - sqrt(6) + 2sqrt(6)) => 3(-sqrt(2) + (-1sqrt(6) + 2sqrt(6)) => 3(-sqrt(2) + sqrt(6))`

Now, rewrite `sqrt(6)` to give:

`3(-sqrt(2) + sqrt(6)) => 3(-sqrt(2) + sqrt(2 * 3)) =>`

`3(-sqrt(2) + (sqrt(2) * sqrt(3))) => 3(-1sqrt(2) + (sqrt(2) * sqrt(3))) =>`

`3((-1+ sqrt(3))sqrt(2)) =>`

`3sqrt(2)(sqrt(3) - 1)`

Answer 2

`3sqrt2color(white)(.)(color(white)(2/2)sqrt(3)-1color(white)(.))`

Explanation:

Always worth looking for common factors and seeing what happens. In roots it is also always looking for squared factors you can 'take outside' the roots.

`18-> 2xx 9 -> 2xx3^2`
`6-> 2 xx 3`
`24->2xx12->2xx3xx4->2xx3xx2^2`

Note that `sqrt(2xx3)` is the same as `sqrt(2)xxsqrt(3)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)(-sqrt18)color(red)(-3sqrt6)color(blue)(+3sqrt24)`

`color(green)(-sqrt(2xx3^2))color(red)(-3sqrt(2xx3)color(blue)(+3sqrt(3xx2xx2^2)`

`color(green)([-3sqrt(2)])color(red)(-[3sqrt(2)xxsqrt(3)]color(blue)(+[6sqrt(3)xxsqrt(2)])`

But `6sqrt3 xx sqrt2` is the same as `3sqrt2xx2sqrt3`
Factoring out `-3sqrt2` gives:

`-3sqrt(2) (1+sqrt3-2sqrt(3))`

`-3sqrt(2)(1-sqrt(3))`

`3sqrt2(sqrt(3)-1)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
this is approximately `3.1058285...`

`-sqrt18-3sqrt6+3sqrt24~~3.1058285....`