Question

How do you simplify `sqrt3 - sqrt27 + 5sqrt12`?

Answer 1

`8sqrt(3)`

Explanation:

`sqrt(3) - sqrt(27) + 5sqrt(12)`
`sqrt(3) - sqrt(9*3)+5sqrt(12)` `color(blue)(" 27 factors into "9*3)`
`sqrt(3)- 3sqrt(3)+5sqrt(12)` `color(blue)(" 9 is a perfect square, so take a 3 out")`
`sqrt(3)-3sqrt(3)+5sqrt(4*3)` `color(blue)(" 12 factors into " 4*3)`
`sqrt(3)-3sqrt(3)+5*2sqrt(3)` `color(blue)(" 4 is a perfect square, so take a 2 out")`
`sqrt(3)-3sqrt(3)+10sqrt(3)` `color(blue)(" To simplify, "5*2=10)`

Now that everything is in like terms of `sqrt(3)`, we can simplify:

`sqrt(3)-3sqrt(3)+10sqrt(3)`
`-2sqrt(3)+10sqrt(3)` `color(blue)(" Subtraction: "1sqrt(3)-3sqrt(3)=-2sqrt(3))`
`8sqrt(3)` `color(blue)(" Addition: "10sqrt(3)+(-2sqrt(3))=8sqrt(3))`

Answer 2

`√3−√27+5√12`

`=8√3`

Explanation:

`√3−√27+5√12`
`=√3−3√3+5√12`
`=√3−3√3+10√3`
`=8√3`

• Simplify each surd to create a 'like' surd, when each number under the root sign is the same. This allows us to calculate the addition of the surds.
• We first simplify √27 to 9√3 = √27 and then simplify the number outside the root sign to = 3 (The square root) this gives us 3√3
• Then we simplify 5√12 to the √12 = 2√3 and then multiply this by 5 = 10√3
• Because each surd is now in the 'like' surd form we can carry out simple addition to complete the equation.
• `=√3−3√3+10√3`
  `=8√3`

Answer 3

`8 sqrt(3)`

Explanation:

Given: `sqrt(3) - sqrt(27) + 5 sqrt(12)`

Simplify using perfect squares and the rule: `sqrt(m*n) = sqrt(m)*sqrt(n)`

Some perfect squares are:
`2^2 = 4`
`3^2 = 9`
`4^2 = 16`
`5^2 = 25`
`6^2 = 36`
...

`sqrt(3) - sqrt(27) + 5 sqrt(12)`

`= sqrt(3) - sqrt(9*3) + 5 sqrt(4*3)`

`= sqrt(3) - sqrt(9)sqrt(3)+ 5 sqrt(4)sqrt(3)`

`= sqrt(3) - 3sqrt(3) + 5 * 2sqrt(3)`

`= sqrt(3) - 3sqrt(3) + 10sqrt(3)`

Since all terms are alike they can be added or subtracted:

`sqrt(3) - sqrt(27) + 5 sqrt(12) = 8 sqrt(3)`