Question

How do you simplify `5sqrt3 + 2sqrt27 + 1/sqrt3`?

Answer 1

`5sqrt3+2sqrt27+1/sqrt3=color(blue)((34sqrt3)/3`

Explanation:

Simplify:

`5sqrt3+2sqrt27+1/sqrt3`

In order to add terms with square roots, the square roots must be the same.

Simplify `2sqrt27` by using prime factorization.

`5sqrt3+2sqrt(3xx3xx3)+1/sqrt3`

`5sqrt3+2sqrt(3^2xx3)+1/sqrt3`

Apply rule: `sqrt(a^2)=a`

`5sqrt3+2xx3sqrt3+1/sqrt3`

Simplify.

`5sqrt3+6sqrt3+1/sqrt3`

Rationalize the denominator in `1/sqrt3` by multiplying by `sqrt3/sqrt3`.

`5sqrt3+6sqrt3+1/sqrt3xxsqrt3/sqrt3`

Apply rule: `sqrtasqrta=a`

`5sqrt3+6sqrt3+sqrt3/3`

In order to add fractions, they must have the same denominator. Multiply the first two terms by `3/3`.

`(3xx5sqrt3)/3 + (3xx6sqrt3)/3 + sqrt3/3`

Simplify.

`(15sqrt3)/3 + (18sqrt3)/3 + sqrt3/3`

Combine all terms.

`(34sqrt3)/3`